GCSE Maths: Inverse functions (Higher)

Key words. The inverse function f⁻¹ reverses f: if f takes 3 to 11, then f⁻¹ takes 11 back to 3. Subject means the letter on its own on one side.

Method. To find the inverse function f⁻¹, write y = f(x), make x the subject, then swap the letters. The inverse undoes what the function did.

1. Write the function as y = … f(x) is the output, which we call y. We want to turn the rule round so that x is given in terms of y.
   y = x + 7
2. Describe what the function does to x, in order. Think of a function machine. To undo it we must reverse every step, in reverse order.
3. Swap the sides so the x part is on the left. We now rearrange to make x the subject.
   x + 7 = y
4. Subtract 7 from both sides. The last thing that happens to the x-part is that 7 is added. The opposite of adding 7 is subtracting 7. Do it to both sides to keep the equation balanced.
   x = y − 7
5. Write x in terms of y. Now x is the subject.
   x = y − 7
6. Swap the letters: write x for the input of the inverse function. A function is written with x as its input, so we rename y as x.
   f⁻¹(x) = x − 7

Check. Test with a number: f(2) = 9. Then f⁻¹(9) = 9 − 7 = 2, which takes us back to the input 2. ✓

Watch out. The notation f⁻¹(x) does not mean 1 divided by f(x). It means the inverse function.

Key words. The inverse function f⁻¹ reverses f: if f takes 3 to 11, then f⁻¹ takes 11 back to 3. Subject means the letter on its own on one side.

Method. To find the inverse function f⁻¹, write y = f(x), make x the subject, then swap the letters. The inverse undoes what the function did.

1. Write the function as y = … f(x) is the output, which we call y. We want to turn the rule round so that x is given in terms of y.
   y = 5x
2. Describe what the function does to x, in order. Think of a function machine. To undo it we must reverse every step, in reverse order.
3. Swap the sides so the x part is on the left. We now rearrange to make x the subject.
   5x = y
4. Divide both sides by 5. The unknown part is being multiplied by 5. The opposite of multiplying by 5 is dividing by 5.
   x = y5
5. Write x in terms of y. Now x is the subject.
   x = y5
6. Swap the letters: write x for the input of the inverse function. A function is written with x as its input, so we rename y as x.
   f⁻¹(x) = x5

Check. Test with a number: f(2) = 10. Then f⁻¹(10) = 105 = 2, which takes us back to the input 2. ✓

Watch out. The notation f⁻¹(x) does not mean 1 divided by f(x). It means the inverse function.

Key words. The inverse function f⁻¹ reverses f: if f takes 3 to 11, then f⁻¹ takes 11 back to 3. Subject means the letter on its own on one side.

Method. To find the inverse function f⁻¹, write y = f(x), make x the subject, then swap the letters. The inverse undoes what the function did.

1. Write the function as y = … f(x) is the output, which we call y. We want to turn the rule round so that x is given in terms of y.
   y = x3
2. Describe what the function does to x, in order. Think of a function machine. To undo it we must reverse every step, in reverse order.
3. Swap the sides so the x part is on the left. We now rearrange to make x the subject.
   x3 = y
4. Multiply both sides by 3. The expression is being divided by 3. The opposite of dividing by 3 is multiplying by 3.
   x = 3y
5. Write x in terms of y. Now x is the subject.
   x = 3y
6. Swap the letters: write x for the input of the inverse function. A function is written with x as its input, so we rename y as x.
   f⁻¹(x) = 3x

Check. Test with a number: f(2) = 23. Then f⁻¹(23) = 3 × 23 = 2, which takes us back to the input 2. ✓

Watch out. The notation f⁻¹(x) does not mean 1 divided by f(x). It means the inverse function.

Key words. The inverse function f⁻¹ reverses f: if f takes 3 to 11, then f⁻¹ takes 11 back to 3. Subject means the letter on its own on one side.

Method. To find the inverse function f⁻¹, write y = f(x), make x the subject, then swap the letters. The inverse undoes what the function did.

1. Write the function as y = … f(x) is the output, which we call y. We want to turn the rule round so that x is given in terms of y.
   y = −x + 10
2. Describe what the function does to x, in order. Think of a function machine. To undo it we must reverse every step, in reverse order.
3. Swap the sides so the x part is on the left. We now rearrange to make x the subject.
   −x + 10 = y
4. Subtract 10 from both sides. The last thing that happens to the x-part is that 10 is added. The opposite of adding 10 is subtracting 10. Do it to both sides to keep the equation balanced.
   −x = y − 10
5. Multiply both sides by −1. The part with the unknown is being multiplied by −1, so multiply by −1 again to undo it (every sign flips).
   x = −y + 10
6. Write x in terms of y. Now x is the subject.
   x = −y + 10
7. Swap the letters: write x for the input of the inverse function. A function is written with x as its input, so we rename y as x.
   f⁻¹(x) = −x + 10

Check. Test with a number: f(2) = 8. Then f⁻¹(8) = −8 + 10 = 2, which takes us back to the input 2. ✓

Watch out. The notation f⁻¹(x) does not mean 1 divided by f(x). It means the inverse function.

Key words. The inverse function f⁻¹ reverses f: if f takes 3 to 11, then f⁻¹ takes 11 back to 3. Subject means the letter on its own on one side.

Method. To find the inverse function f⁻¹, write y = f(x), make x the subject, then swap the letters. The inverse undoes what the function did.

1. Write the function as y = … f(x) is the output, which we call y. We want to turn the rule round so that x is given in terms of y.
   y = 2x + 3
2. Describe what the function does to x, in order. Think of a function machine. To undo it we must reverse every step, in reverse order.
3. Swap the sides so the x part is on the left. We now rearrange to make x the subject.
   2x + 3 = y
4. Subtract 3 from both sides. The last thing that happens to the x-part is that 3 is added. The opposite of adding 3 is subtracting 3. Do it to both sides to keep the equation balanced.
   2x = y − 3
5. Divide both sides by 2. The unknown part is being multiplied by 2. The opposite of multiplying by 2 is dividing by 2.
   x = y − 32
6. Write x in terms of y. Now x is the subject.
   x = y − 32
7. Swap the letters: write x for the input of the inverse function. A function is written with x as its input, so we rename y as x.
   f⁻¹(x) = x − 32

Check. Test with a number: f(2) = 7. Then f⁻¹(7) = 72 − 32 = 2, which takes us back to the input 2. ✓

Watch out. The notation f⁻¹(x) does not mean 1 divided by f(x). It means the inverse function.

Key words. The inverse function f⁻¹ reverses f: if f takes 3 to 11, then f⁻¹ takes 11 back to 3. Subject means the letter on its own on one side.

Method. To find the inverse function f⁻¹, write y = f(x), make x the subject, then swap the letters. The inverse undoes what the function did.

1. Write the function as y = … f(x) is the output, which we call y. We want to turn the rule round so that x is given in terms of y.
   y = 3x − 4
2. Describe what the function does to x, in order. Think of a function machine. To undo it we must reverse every step, in reverse order.
3. Swap the sides so the x part is on the left. We now rearrange to make x the subject.
   3x − 4 = y
4. Add 4 to both sides. The last thing that happens is that 4 is subtracted. The opposite of subtracting 4 is adding 4. Do it to both sides.
   3x = y + 4
5. Divide both sides by 3. The unknown part is being multiplied by 3. The opposite of multiplying by 3 is dividing by 3.
   x = y + 43
6. Write x in terms of y. Now x is the subject.
   x = y + 43
7. Swap the letters: write x for the input of the inverse function. A function is written with x as its input, so we rename y as x.
   f⁻¹(x) = x + 43

Check. Test with a number: f(2) = 2. Then f⁻¹(2) = 23 + 43 = 2, which takes us back to the input 2. ✓

Watch out. The notation f⁻¹(x) does not mean 1 divided by f(x). It means the inverse function.

Key words. The inverse function f⁻¹ reverses f: if f takes 3 to 11, then f⁻¹ takes 11 back to 3. Subject means the letter on its own on one side.

Method. To find the inverse function f⁻¹, write y = f(x), make x the subject, then swap the letters. The inverse undoes what the function did.

1. Write the function as y = … f(x) is the output, which we call y. We want to turn the rule round so that x is given in terms of y.
   y = x − 25
2. Describe what the function does to x, in order. Think of a function machine. To undo it we must reverse every step, in reverse order.
3. Swap the sides so the x part is on the left. We now rearrange to make x the subject.
   x − 25 = y
4. Multiply both sides by 5. The expression is being divided by 5. The opposite of dividing by 5 is multiplying by 5.
   x − 2 = 5y
5. Add 2 to both sides. The last thing that happens is that 2 is subtracted. The opposite of subtracting 2 is adding 2. Do it to both sides.
   x = 5y + 2
6. Write x in terms of y. Now x is the subject.
   x = 5y + 2
7. Swap the letters: write x for the input of the inverse function. A function is written with x as its input, so we rename y as x.
   f⁻¹(x) = 5x + 2

Check. Test with a number: f(2) = 0. Then f⁻¹(0) = 5 × 0 + 2 = 2, which takes us back to the input 2. ✓

Watch out. The notation f⁻¹(x) does not mean 1 divided by f(x). It means the inverse function.

Key words. The inverse function f⁻¹ reverses f: if f takes 3 to 11, then f⁻¹ takes 11 back to 3. Subject means the letter on its own on one side.

Method. To find the inverse function f⁻¹, write y = f(x), make x the subject, then swap the letters. The inverse undoes what the function did.

1. Write the function as y = … f(x) is the output, which we call y. We want to turn the rule round so that x is given in terms of y.
   y = 4(x + 1)
2. Describe what the function does to x, in order. Think of a function machine. To undo it we must reverse every step, in reverse order.
3. Swap the sides so the x part is on the left. We now rearrange to make x the subject.
   4(x + 1) = y
4. Divide both sides by 4. The unknown part is being multiplied by 4. The opposite of multiplying by 4 is dividing by 4.
   x + 1 = y4
5. Subtract 1 from both sides. The last thing that happens to the x-part is that 1 is added. The opposite of adding 1 is subtracting 1. Do it to both sides to keep the equation balanced.
   x = y − 44
6. Write x in terms of y. Now x is the subject.
   x = y − 44
7. Swap the letters: write x for the input of the inverse function. A function is written with x as its input, so we rename y as x.
   f⁻¹(x) = x − 44

Check. Test with a number: f(2) = 12. Then f⁻¹(12) = 124 − 1 = 2, which takes us back to the input 2. ✓

Watch out. The notation f⁻¹(x) does not mean 1 divided by f(x). It means the inverse function.

Method. To find f⁻¹(7) without finding the whole inverse function, solve f(x) = 7.

1. Understand what is being asked. f⁻¹(7) is the number that the function f turns into 7. So it is the input x for which the output f(x) equals 7.
   f⁻¹(7) = the value of x that makes f(x) = 7
2. Write the equation. We are told that the output is 7, so f(x) = 7. Put the rule in for f(x).
   4x − 1 = 7
3. Add 1 to both sides. The last thing that happens is that 1 is subtracted. The opposite of subtracting 1 is adding 1. Do it to both sides.
   4x = 8
4. Divide both sides by 4. The unknown part is being multiplied by 4. The opposite of multiplying by 4 is dividing by 4.
   x = 2
5. Write the answer. x is now on its own.
   x = 2

Check. Test it: f(2) = 4 × 2 − 1 = 7, so f sends 2 to 7, and the inverse sends 7 back to 2. ✓

Watch out. It is easy to work out f(7) by mistake. f⁻¹(7) asks for the input that gives 7.

Method. To find f⁻¹(−5) without finding the whole inverse function, solve f(x) = −5.

1. Understand what is being asked. f⁻¹(−5) is the number that the function f turns into −5. So it is the input x for which the output f(x) equals −5.
   f⁻¹(−5) = the value of x that makes f(x) = −5
2. Write the equation. We are told that the output is −5, so f(x) = −5. Put the rule in for f(x).
   4x − 1 = −5
3. Add 1 to both sides. The last thing that happens is that 1 is subtracted. The opposite of subtracting 1 is adding 1. Do it to both sides.
   4x = −4
4. Divide both sides by 4. The unknown part is being multiplied by 4. The opposite of multiplying by 4 is dividing by 4.
   x = −1
5. Write the answer. x is now on its own.
   x = −1

Check. Test it: f(−1) = 4 × (−1) − 1 = −5, so f sends −1 to −5, and the inverse sends −5 back to −1. ✓

Watch out. It is easy to work out f(7) by mistake. f⁻¹(7) asks for the input that gives 7.

Method. To find f⁻¹(0) without finding the whole inverse function, solve f(x) = 0.

1. Understand what is being asked. f⁻¹(0) is the number that the function f turns into 0. So it is the input x for which the output f(x) equals 0.
   f⁻¹(0) = the value of x that makes f(x) = 0
2. Write the equation. We are told that the output is 0, so f(x) = 0. Put the rule in for f(x).
   4x − 1 = 0
3. Add 1 to both sides. The last thing that happens is that 1 is subtracted. The opposite of subtracting 1 is adding 1. Do it to both sides.
   4x = 1
4. Divide both sides by 4. The unknown part is being multiplied by 4. The opposite of multiplying by 4 is dividing by 4.
   x = 14
5. Write the answer. x is now on its own.
   x = 0.25

Check. Test it: f(0.25) = 4 × 0.25 − 1 = 0, so f sends 0.25 to 0, and the inverse sends 0 back to 0.25. ✓

Watch out. It is easy to work out f(7) by mistake. f⁻¹(7) asks for the input that gives 7.

Key words. The inverse function f⁻¹ reverses f: if f takes 3 to 11, then f⁻¹ takes 11 back to 3. Subject means the letter on its own on one side.

Method. To find the inverse function f⁻¹, write y = f(x), make x the subject, then swap the letters. The inverse undoes what the function did.

1. Write the function as y = … f(x) is the output, which we call y. We want to turn the rule round so that x is given in terms of y.
   y = 8x
2. Describe what the function does to x, in order. Think of a function machine. To undo it we must reverse every step, in reverse order.
3. Swap the sides so the x part is on the left. We now rearrange to make x the subject.
   8x = y
4. The unknown is on the bottom of the fraction: multiply both sides by it, then divide both sides by the other side. Multiplying both sides by x removes the fraction. Then dividing both sides by y leaves x on its own.
   8 = yx
   8y = x
5. Write x in terms of y. Now x is the subject.
   x = 8y
6. Swap the letters: write x for the input of the inverse function. A function is written with x as its input, so we rename y as x.
   f⁻¹(x) = 8x

Check. Test with a number: f(2) = 4. Then f⁻¹(4) = 84 = 2, which takes us back to the input 2. ✓

Watch out. The notation f⁻¹(x) does not mean 1 divided by f(x). It means the inverse function.

Key words. The inverse function f⁻¹ reverses f: if f takes 3 to 11, then f⁻¹ takes 11 back to 3. Subject means the letter on its own on one side.

Method. To find the inverse function f⁻¹, write y = f(x), make x the subject, then swap the letters. The inverse undoes what the function did.

1. Write the function as y = … f(x) is the output, which we call y. We want to turn the rule round so that x is given in terms of y.
   y = 6x − 2
2. Describe what the function does to x, in order. Think of a function machine. To undo it we must reverse every step, in reverse order.
3. Swap the sides so the x part is on the left. We now rearrange to make x the subject.
   6x − 2 = y
4. Add 2 to both sides. The last thing that happens is that 2 is subtracted. The opposite of subtracting 2 is adding 2. Do it to both sides.
   6x = y + 2
5. The unknown is on the bottom of the fraction: multiply both sides by it, then divide both sides by the other side. Multiplying both sides by x removes the fraction. Then dividing both sides by y + 2 leaves x on its own.
   6 = y + 2x
   6y + 2 = x
6. Write x in terms of y. Now x is the subject.
   x = 6y + 2
7. Swap the letters: write x for the input of the inverse function. A function is written with x as its input, so we rename y as x.
   f⁻¹(x) = 6x + 2

Check. Test with a number: f(2) = 1. Then f⁻¹(1) = 61 + 2 = 2, which takes us back to the input 2. ✓

Watch out. The notation f⁻¹(x) does not mean 1 divided by f(x). It means the inverse function.

Key words. The inverse function f⁻¹ reverses f: if f takes 3 to 11, then f⁻¹ takes 11 back to 3. Subject means the letter on its own on one side.

Method. To find the inverse function f⁻¹, write y = f(x), make x the subject, then swap the letters. The inverse undoes what the function did.

1. Write the function as y = … f(x) is the output, which we call y. We want to turn the rule round so that x is given in terms of y.
   y = 4x + 1
2. Describe what the function does to x, in order. Think of a function machine. To undo it we must reverse every step, in reverse order.
3. Swap the sides so the x part is on the left. We now rearrange to make x the subject.
   4x + 1 = y
4. The unknown is on the bottom of the fraction: multiply both sides by it, then divide both sides by the other side. Multiplying both sides by x + 1 removes the fraction. Then dividing both sides by y leaves x + 1 on its own.
   4 = y(x + 1)
   4y = x + 1
5. Subtract 1 from both sides. The last thing that happens to the x-part is that 1 is added. The opposite of adding 1 is subtracting 1. Do it to both sides to keep the equation balanced.
   x = −y + 4y
6. Write x in terms of y. Now x is the subject.
   x = −y + 4y
7. Swap the letters: write x for the input of the inverse function. A function is written with x as its input, so we rename y as x.
   f⁻¹(x) = −x + 4x

Check. Test with a number: f(2) = 43. Then f⁻¹(43) = 443 − 1 = 2, which takes us back to the input 2. ✓

Watch out. The notation f⁻¹(x) does not mean 1 divided by f(x). It means the inverse function.

Key words. The inverse function f⁻¹ reverses f: if f takes 3 to 11, then f⁻¹ takes 11 back to 3. Subject means the letter on its own on one side.

Method. To find the inverse function f⁻¹, write y = f(x), make x the subject, then swap the letters. The inverse undoes what the function did.

1. Write the function as y = … f(x) is the output, which we call y. We want to turn the rule round so that x is given in terms of y.
   y = 32x
2. Describe what the function does to x, in order. Think of a function machine. To undo it we must reverse every step, in reverse order.
3. Swap the sides so the x part is on the left. We now rearrange to make x the subject.
   32x = y
4. The unknown is on the bottom of the fraction: multiply both sides by it, then divide both sides by the other side. Multiplying both sides by x removes the fraction. Then dividing both sides by y leaves x on its own.
   32 = yx
   32y = x
5. Write x in terms of y. Now x is the subject.
   x = 32y
6. Swap the letters: write x for the input of the inverse function. A function is written with x as its input, so we rename y as x.
   f⁻¹(x) = 32x

Check. Test with a number: f(2) = 0.75. Then f⁻¹(0.75) = 32 × 0.75 = 2, which takes us back to the input 2. ✓

Watch out. The notation f⁻¹(x) does not mean 1 divided by f(x). It means the inverse function.

Key words. The inverse function f⁻¹ reverses f: if f takes 3 to 11, then f⁻¹ takes 11 back to 3. Subject means the letter on its own on one side.

Method. To find the inverse function f⁻¹, write y = f(x), make x the subject, then swap the letters. The inverse undoes what the function did.

1. Write the function as y = … f(x) is the output, which we call y. We want to turn the rule round so that x is given in terms of y.
   y = x² + 3
2. Describe what the function does to x, in order. Think of a function machine. To undo it we must reverse every step, in reverse order.
3. Swap the sides so the x part is on the left. We now rearrange to make x the subject.
   x² + 3 = y
4. Subtract 3 from both sides. The last thing that happens to the x-part is that 3 is added. The opposite of adding 3 is subtracting 3. Do it to both sides to keep the equation balanced.
   x² = y − 3
5. Take the square root of both sides. The opposite of squaring is the square root. (Because the function is only used for x ≥ 0, we take the positive root.)
   x = √(y − 3)
6. Write x in terms of y. Now x is the subject.
   x = √(y − 3)
7. Swap the letters: write x for the input of the inverse function. A function is written with x as its input, so we rename y as x.
   f⁻¹(x) = √(x − 3)

Check. Test with a number: f(2) = 7. Then f⁻¹(7) = √(7 − 3) = 2, which takes us back to the input 2. ✓

Watch out. The notation f⁻¹(x) does not mean 1 divided by f(x). It means the inverse function.

Key words. The inverse function f⁻¹ reverses f: if f takes 3 to 11, then f⁻¹ takes 11 back to 3. Subject means the letter on its own on one side.

Method. To find the inverse function f⁻¹, write y = f(x), make x the subject, then swap the letters. The inverse undoes what the function did.

1. Write the function as y = … f(x) is the output, which we call y. We want to turn the rule round so that x is given in terms of y.
   y = (x − 1)²
2. Describe what the function does to x, in order. Think of a function machine. To undo it we must reverse every step, in reverse order.
3. Swap the sides so the x part is on the left. We now rearrange to make x the subject.
   (x − 1)² = y
4. Take the square root of both sides. The opposite of squaring is the square root. (Because the function is only used for x − 1 ≥ 0, we take the positive root.)
   x − 1 = √y
5. Add 1 to both sides. The last thing that happens is that 1 is subtracted. The opposite of subtracting 1 is adding 1. Do it to both sides.
   x = √y + 1
6. Write x in terms of y. Now x is the subject.
   x = √y + 1
7. Swap the letters: write x for the input of the inverse function. A function is written with x as its input, so we rename y as x.
   f⁻¹(x) = √x + 1

Check. Test with a number: f(2) = 1. Then f⁻¹(1) = √1 + 1 = 2, which takes us back to the input 2. ✓

Watch out. The notation f⁻¹(x) does not mean 1 divided by f(x). It means the inverse function.

Key words. The inverse function f⁻¹ reverses f: if f takes 3 to 11, then f⁻¹ takes 11 back to 3. Subject means the letter on its own on one side.

Method. To find the inverse function f⁻¹, write y = f(x), make x the subject, then swap the letters. The inverse undoes what the function did.

1. Write the function as y = … f(x) is the output, which we call y. We want to turn the rule round so that x is given in terms of y.
   y = √(x + 4)
2. Describe what the function does to x, in order. Think of a function machine. To undo it we must reverse every step, in reverse order.
3. Swap the sides so the x part is on the left. We now rearrange to make x the subject.
   √(x + 4) = y
4. Square both sides. The opposite of a square root is squaring.
   x + 4 = y²
5. Subtract 4 from both sides. The last thing that happens to the x-part is that 4 is added. The opposite of adding 4 is subtracting 4. Do it to both sides to keep the equation balanced.
   x = y² − 4
6. Write x in terms of y. Now x is the subject.
   x = y² − 4
7. Swap the letters: write x for the input of the inverse function. A function is written with x as its input, so we rename y as x.
   f⁻¹(x) = x² − 4

Check. Test with a number: f(1) = 527759236021. Then f⁻¹(527759236021) = 527759236021² − 4 = 1, which takes us back to the input 1. ✓

Watch out. The notation f⁻¹(x) does not mean 1 divided by f(x). It means the inverse function.

Key words. The inverse function f⁻¹ reverses f: if f takes 3 to 11, then f⁻¹ takes 11 back to 3. Subject means the letter on its own on one side.

Method. To find the inverse function f⁻¹, write y = f(x), make x the subject, then swap the letters. The inverse undoes what the function did.

1. Write the function as y = … f(x) is the output, which we call y. We want to turn the rule round so that x is given in terms of y.
   y = x³ − 1
2. Describe what the function does to x, in order. Think of a function machine. To undo it we must reverse every step, in reverse order.
3. Swap the sides so the x part is on the left. We now rearrange to make x the subject.
   x³ − 1 = y
4. Add 1 to both sides. The last thing that happens is that 1 is subtracted. The opposite of subtracting 1 is adding 1. Do it to both sides.
   x³ = y + 1
5. Take the cube root of both sides. The opposite of cubing is the cube root.
   x = ∛(y + 1)
6. Write x in terms of y. Now x is the subject.
   x = ∛(y + 1)
7. Swap the letters: write x for the input of the inverse function. A function is written with x as its input, so we rename y as x.
   f⁻¹(x) = ∛(x + 1)

Check. Test with a number: f(2) = 7. Then f⁻¹(7) = ∛(7 + 1) = 2, which takes us back to the input 2. ✓

Watch out. The notation f⁻¹(x) does not mean 1 divided by f(x). It means the inverse function.

Key words. The inverse function f⁻¹ reverses f: if f takes 3 to 11, then f⁻¹ takes 11 back to 3. Subject means the letter on its own on one side.

Method. To find the inverse function f⁻¹, write y = f(x), make x the subject, then swap the letters. The inverse undoes what the function did.

1. Write the function as y = … f(x) is the output, which we call y. We want to turn the rule round so that x is given in terms of y.
   y = 3(x − 2) + 1
2. Describe what the function does to x, in order. Think of a function machine. To undo it we must reverse every step, in reverse order.
3. Swap the sides so the x part is on the left. We now rearrange to make x the subject.
   3(x − 2) + 1 = y
4. Subtract 1 from both sides. The last thing that happens to the x-part is that 1 is added. The opposite of adding 1 is subtracting 1. Do it to both sides to keep the equation balanced.
   3(x − 2) = y − 1
5. Divide both sides by 3. The unknown part is being multiplied by 3. The opposite of multiplying by 3 is dividing by 3.
   x − 2 = y − 13
6. Add 2 to both sides. The last thing that happens is that 2 is subtracted. The opposite of subtracting 2 is adding 2. Do it to both sides.
   x = y + 53
7. Write x in terms of y. Now x is the subject.
   x = y + 53
8. Swap the letters: write x for the input of the inverse function. A function is written with x as its input, so we rename y as x.
   f⁻¹(x) = x + 53

Check. Test with a number: f(2) = 1. Then f⁻¹(1) = 13 + 53 = 2, which takes us back to the input 2. ✓

Watch out. The notation f⁻¹(x) does not mean 1 divided by f(x). It means the inverse function.

Key words. The inverse function f⁻¹ reverses f: if f takes 3 to 11, then f⁻¹ takes 11 back to 3. Subject means the letter on its own on one side.

Method. To find the inverse function f⁻¹, write y = f(x), make x the subject, then swap the letters. The inverse undoes what the function did.

1. Write the function as y = … f(x) is the output, which we call y. We want to turn the rule round so that x is given in terms of y.
   y = −2x + 5
2. Describe what the function does to x, in order. Think of a function machine. To undo it we must reverse every step, in reverse order.
3. Swap the sides so the x part is on the left. We now rearrange to make x the subject.
   −2x + 5 = y
4. Subtract 5 from both sides. The last thing that happens to the x-part is that 5 is added. The opposite of adding 5 is subtracting 5. Do it to both sides to keep the equation balanced.
   −2x = y − 5
5. Divide both sides by −2. The unknown part is being multiplied by −2. The opposite of multiplying by −2 is dividing by −2.
   x = −y + 52
6. Write x in terms of y. Now x is the subject.
   x = −y + 52
7. Swap the letters: write x for the input of the inverse function. A function is written with x as its input, so we rename y as x.
   f⁻¹(x) = −x + 52

Check. Test with a number: f(2) = 1. Then f⁻¹(1) = −12 + 52 = 2, which takes us back to the input 2. ✓

Watch out. The notation f⁻¹(x) does not mean 1 divided by f(x). It means the inverse function.

Key words. The inverse function f⁻¹ reverses f: if f takes 3 to 11, then f⁻¹ takes 11 back to 3. Subject means the letter on its own on one side.

Method. To find the inverse function f⁻¹, write y = f(x), make x the subject, then swap the letters. The inverse undoes what the function did.

1. Write the function as y = … f(x) is the output, which we call y. We want to turn the rule round so that x is given in terms of y.
   y = (x + 2)³
2. Describe what the function does to x, in order. Think of a function machine. To undo it we must reverse every step, in reverse order.
3. Swap the sides so the x part is on the left. We now rearrange to make x the subject.
   (x + 2)³ = y
4. Take the cube root of both sides. The opposite of cubing is the cube root.
   x + 2 = ∛y
5. Subtract 2 from both sides. The last thing that happens to the x-part is that 2 is added. The opposite of adding 2 is subtracting 2. Do it to both sides to keep the equation balanced.
   x = ∛y − 2
6. Write x in terms of y. Now x is the subject.
   x = ∛y − 2
7. Swap the letters: write x for the input of the inverse function. A function is written with x as its input, so we rename y as x.
   f⁻¹(x) = ∛x − 2

Check. Test with a number: f(2) = 64. Then f⁻¹(64) = ∛64 − 2 = 2, which takes us back to the input 2. ✓

Watch out. The notation f⁻¹(x) does not mean 1 divided by f(x). It means the inverse function.

Key words. The inverse function f⁻¹ reverses f: if f takes 3 to 11, then f⁻¹ takes 11 back to 3. Subject means the letter on its own on one side.

Method. To find the inverse function f⁻¹, write y = f(x), make x the subject, then swap the letters. The inverse undoes what the function did.

1. Write the function as y = … f(x) is the output, which we call y. We want to turn the rule round so that x is given in terms of y.
   y = x + 42
2. Describe what the function does to x, in order. Think of a function machine. To undo it we must reverse every step, in reverse order.
3. Swap the sides so the x part is on the left. We now rearrange to make x the subject.
   x + 42 = y
4. Multiply both sides by 2. The expression is being divided by 2. The opposite of dividing by 2 is multiplying by 2.
   x + 4 = 2y
5. Subtract 4 from both sides. The last thing that happens to the x-part is that 4 is added. The opposite of adding 4 is subtracting 4. Do it to both sides to keep the equation balanced.
   x = 2y − 4
6. Write x in terms of y. Now x is the subject.
   x = 2y − 4
7. Swap the letters: write x for the input of the inverse function. A function is written with x as its input, so we rename y as x.
   f⁻¹(x) = 2x − 4

Check. Test with a number: f(2) = 3. Then f⁻¹(3) = 2 × 3 − 4 = 2, which takes us back to the input 2. ✓

Watch out. The notation f⁻¹(x) does not mean 1 divided by f(x). It means the inverse function.

Method. Rearrange y = f(x) to make x the subject. When x appears twice, multiply out the fraction, collect the x terms and factorise.

1. Write the function as y = … This time x appears twice (top and bottom), so we cannot simply undo operations one by one. We will rearrange carefully.
   y = 3x + 1x − 2
2. Multiply both sides by the bottom. This removes the fraction.
   y(x − 2) = 3x + 1
3. Expand the bracket. Multiply y by each term inside.
   xy − 2y = 3x + 1
4. Get all the x terms on one side and everything else on the other. We need x on its own eventually, so collect the terms containing x together.
   xy − 3x = 2y + 1
5. Factorise the left side. Take x out as a common factor so that x appears only once.
   x(y − 3) = 2y + 1
6. Divide both sides by the bracket. This leaves x on its own.
   x = 2y + 1y − 3
7. Swap the letters. Write x for the input of the inverse function.
   f⁻¹(x) = 2x + 1x − 3

Check. Test: f(3) = 10, and f⁻¹(10) = 3, the number we started with. ✓

Watch out. Do not forget that y multiplies both terms of the bracket when you expand.

Method. Rearrange y = f(x) to make x the subject. When x appears twice, multiply out the fraction, collect the x terms and factorise.

1. Write the function as y = … This time x appears twice (top and bottom), so we cannot simply undo operations one by one. We will rearrange carefully.
   y = 2x − 1x + 3
2. Multiply both sides by the bottom. This removes the fraction.
   y(x + 3) = 2x − 1
3. Expand the bracket. Multiply y by each term inside.
   xy + 3y = 2x − 1
4. Get all the x terms on one side and everything else on the other. We need x on its own eventually, so collect the terms containing x together.
   xy − 2x = −3y − 1
5. Factorise the left side. Take x out as a common factor so that x appears only once.
   x(y − 2) = −3y − 1
6. Divide both sides by the bracket. This leaves x on its own.
   x = −3y − 1y − 2
7. Swap the letters. Write x for the input of the inverse function.
   f⁻¹(x) = −3x − 1x − 2

Check. Test: f(3) = 56, and f⁻¹(56) = 3, the number we started with. ✓

Watch out. Do not forget that y multiplies both terms of the bracket when you expand.

Method. Rearrange y = f(x) to make x the subject. When x appears twice, multiply out the fraction, collect the x terms and factorise.

1. Write the function as y = … This time x appears twice (top and bottom), so we cannot simply undo operations one by one. We will rearrange carefully.
   y = xx + 1
2. Multiply both sides by the bottom. This removes the fraction.
   y(x + 1) = x
3. Expand the bracket. Multiply y by each term inside.
   xy + y = x
4. Get all the x terms on one side and everything else on the other. We need x on its own eventually, so collect the terms containing x together.
   xy − x = −y
5. Factorise the left side. Take x out as a common factor so that x appears only once.
   x(y − 1) = −y
6. Divide both sides by the bracket. This leaves x on its own.
   x = −yy − 1
7. Swap the letters. Write x for the input of the inverse function.
   f⁻¹(x) = −xx − 1

Check. Test: f(3) = 0.75, and f⁻¹(0.75) = 3, the number we started with. ✓

Watch out. Do not forget that y multiplies both terms of the bracket when you expand.

Method. The two graphs are mirror images of each other in one line. Find the line through points that have equal x and y coordinates.

1. Look at the dashed line. It passes through (0, 0), (1, 1), (2, 2), (−3, −3).
   every point has x = y
2. Write the equation.
   y = x

Check. The point (1, 3) on the blue graph reflects to (3, 1) on the orange graph, and the line half-way between is y = x. ✓

Method. Read the blue graph at x = 2. An inverse swaps inputs and outputs.

1. Read f(2). The blue line passes through (2, 5).
   f(2) = 5
2. Swap input and output. Since f turns 2 into 5, the inverse turns 5 back into 2. The orange line passes through (5, 2).
   f⁻¹(5) = 2

Check. Check with the rule: 2 × 2 + 1 = 5. ✓

Key words. The inverse function f⁻¹ reverses f: if f takes 3 to 11, then f⁻¹ takes 11 back to 3. Subject means the letter on its own on one side.

Method. To find the inverse function f⁻¹, write y = f(x), make x the subject, then swap the letters. The inverse undoes what the function did.

1. Write the function as y = … f(x) is the output, which we call y. We want to turn the rule round so that x is given in terms of y.
   y = 2x + 1
2. Describe what the function does to x, in order. Think of a function machine. To undo it we must reverse every step, in reverse order.
3. Swap the sides so the x part is on the left. We now rearrange to make x the subject.
   2x + 1 = y
4. Subtract 1 from both sides. The last thing that happens to the x-part is that 1 is added. The opposite of adding 1 is subtracting 1. Do it to both sides to keep the equation balanced.
   2x = y − 1
5. Divide both sides by 2. The unknown part is being multiplied by 2. The opposite of multiplying by 2 is dividing by 2.
   x = y − 12
6. Write x in terms of y. Now x is the subject.
   x = y − 12
7. Swap the letters: write x for the input of the inverse function. A function is written with x as its input, so we rename y as x.
   f⁻¹(x) = x − 12

Check. Test with a number: f(2) = 5. Then f⁻¹(5) = 52 − 12 = 2, which takes us back to the input 2. ✓

Watch out. The notation f⁻¹(x) does not mean 1 divided by f(x). It means the inverse function.