GCSE Maths: Function notation (Higher)

Key words. A function is a rule that turns each input into one output. f(x) is read “f of x” and means the output when the input is x.

Method. To find f(a), substitute the number a for x everywhere in the rule, then work out the answer.

1. Write down the function. f(x) is a rule that says what to do to the input x. The letter x is a placeholder for whichever number we put in.
   f(x) = 4x − 3
2. Replace every x with 2. The number inside the bracket of f(2) is the input, so it goes where the x was.
   f(2) = 4 × 2 − 3
3. Work out each part separately (powers first, then multiplying). BIDMAS: indices before multiplication, then add or subtract.
   = 8 − 3
4. Add and subtract. Work from left to right.
   = 5

Check. Work it out again on a calculator, typing the brackets exactly as shown: the result is 5. ✓

Watch out. Put brackets round negative inputs. For example (−3)² = 9 but −3² = −9.

Key words. A function is a rule that turns each input into one output. f(x) is read “f of x” and means the output when the input is x.

Method. To find f(a), substitute the number a for x everywhere in the rule, then work out the answer.

1. Write down the function. f(x) is a rule that says what to do to the input x. The letter x is a placeholder for whichever number we put in.
   f(x) = 4x − 3
2. Replace every x with 0. The number inside the bracket of f(0) is the input, so it goes where the x was.
   f(0) = 4 × 0 − 3
3. Work out each part separately (powers first, then multiplying). BIDMAS: indices before multiplication, then add or subtract.
   = 0 − 3
4. Add and subtract. Work from left to right.
   = −3

Check. Work it out again on a calculator, typing the brackets exactly as shown: the result is −3. ✓

Watch out. Put brackets round negative inputs. For example (−3)² = 9 but −3² = −9.

Key words. A function is a rule that turns each input into one output. f(x) is read “f of x” and means the output when the input is x.

Method. To find f(a), substitute the number a for x everywhere in the rule, then work out the answer.

1. Write down the function. f(x) is a rule that says what to do to the input x. The letter x is a placeholder for whichever number we put in.
   f(x) = 4x − 3
2. Replace every x with −3, using brackets. The number inside the bracket of f(−3) is the input, so it goes where the x was. Brackets matter with negative numbers.
   f(−3) = 4 × (−3) − 3
3. Work out each part separately (powers first, then multiplying). BIDMAS: indices before multiplication, then add or subtract.
   = −12 − 3
4. Add and subtract. Work from left to right.
   = −15

Check. Work it out again on a calculator, typing the brackets exactly as shown: the result is −15. ✓

Watch out. Put brackets round negative inputs. For example (−3)² = 9 but −3² = −9.

Key words. A function is a rule that turns each input into one output. f(x) is read “f of x” and means the output when the input is x.

Method. To find f(a), substitute the number a for x everywhere in the rule, then work out the answer.

1. Write down the function. f(x) is a rule that says what to do to the input x. The letter x is a placeholder for whichever number we put in.
   f(x) = 4x − 3
2. Replace every x with 10. The number inside the bracket of f(10) is the input, so it goes where the x was.
   f(10) = 4 × 10 − 3
3. Work out each part separately (powers first, then multiplying). BIDMAS: indices before multiplication, then add or subtract.
   = 40 − 3
4. Add and subtract. Work from left to right.
   = 37

Check. Work it out again on a calculator, typing the brackets exactly as shown: the result is 37. ✓

Watch out. Put brackets round negative inputs. For example (−3)² = 9 but −3² = −9.

Key words. A function is a rule that turns each input into one output. f(x) is read “f of x” and means the output when the input is x.

Method. To find f(a), substitute the number a for x everywhere in the rule, then work out the answer.

1. Write down the function. g(x) is a rule that says what to do to the input x. The letter x is a placeholder for whichever number we put in.
   g(x) = x² + 1
2. Replace every x with 3. The number inside the bracket of g(3) is the input, so it goes where the x was.
   g(3) = 3² + 1
3. Work out each part separately (powers first, then multiplying). BIDMAS: indices before multiplication, then add or subtract.
   = 9 + 1
4. Add and subtract. Work from left to right.
   = 10

Check. Work it out again on a calculator, typing the brackets exactly as shown: the result is 10. ✓

Watch out. Put brackets round negative inputs. For example (−3)² = 9 but −3² = −9.

Key words. A function is a rule that turns each input into one output. f(x) is read “f of x” and means the output when the input is x.

Method. To find f(a), substitute the number a for x everywhere in the rule, then work out the answer.

1. Write down the function. g(x) is a rule that says what to do to the input x. The letter x is a placeholder for whichever number we put in.
   g(x) = x² + 1
2. Replace every x with −4, using brackets. The number inside the bracket of g(−4) is the input, so it goes where the x was. Brackets matter with negative numbers.
   g(−4) = (−4)² + 1
3. Work out each part separately (powers first, then multiplying). BIDMAS: indices before multiplication, then add or subtract.
   = 16 + 1
4. Add and subtract. Work from left to right.
   = 17

Check. Work it out again on a calculator, typing the brackets exactly as shown: the result is 17. ✓

Watch out. Put brackets round negative inputs. For example (−3)² = 9 but −3² = −9.

Key words. A function is a rule that turns each input into one output. f(x) is read “f of x” and means the output when the input is x.

Method. To find f(a), substitute the number a for x everywhere in the rule, then work out the answer.

1. Write down the function. g(x) is a rule that says what to do to the input x. The letter x is a placeholder for whichever number we put in.
   g(x) = x² + 1
2. Replace every x with 0. The number inside the bracket of g(0) is the input, so it goes where the x was.
   g(0) = 0² + 1
3. Work out each part separately (powers first, then multiplying). BIDMAS: indices before multiplication, then add or subtract.
   = 0 + 1
4. Add and subtract. Work from left to right.
   = 1

Check. Work it out again on a calculator, typing the brackets exactly as shown: the result is 1. ✓

Watch out. Put brackets round negative inputs. For example (−3)² = 9 but −3² = −9.

Key words. A function is a rule that turns each input into one output. f(x) is read “f of x” and means the output when the input is x.

Method. To find f(a), substitute the number a for x everywhere in the rule, then work out the answer.

1. Write down the function. g(x) is a rule that says what to do to the input x. The letter x is a placeholder for whichever number we put in.
   g(x) = x² + 1
2. Replace every x with 0.5. The number inside the bracket of g(0.5) is the input, so it goes where the x was.
   g(0.5) = 0.5² + 1
3. Work out each part separately (powers first, then multiplying). BIDMAS: indices before multiplication, then add or subtract.
   = 0.25 + 1
4. Add and subtract. Work from left to right.
   = 1.25

Check. Work it out again on a calculator, typing the brackets exactly as shown: the result is 1.25. ✓

Watch out. Put brackets round negative inputs. For example (−3)² = 9 but −3² = −9.

Key words. A function is a rule that turns each input into one output. f(x) is read “f of x” and means the output when the input is x.

Method. To find f(a), substitute the number a for x everywhere in the rule, then work out the answer.

1. Write down the function. h(x) is a rule that says what to do to the input x. The letter x is a placeholder for whichever number we put in.
   h(x) = 20x
2. Replace every x with 4. The number inside the bracket of h(4) is the input, so it goes where the x was.
   h(4) = 204
3. Work it out. Follow BIDMAS.
   = 5

Check. Work it out again on a calculator, typing the brackets exactly as shown: the result is 5. ✓

Watch out. Put brackets round negative inputs. For example (−3)² = 9 but −3² = −9.

Key words. A function is a rule that turns each input into one output. f(x) is read “f of x” and means the output when the input is x.

Method. To find f(a), substitute the number a for x everywhere in the rule, then work out the answer.

1. Write down the function. h(x) is a rule that says what to do to the input x. The letter x is a placeholder for whichever number we put in.
   h(x) = 20x
2. Replace every x with −5, using brackets. The number inside the bracket of h(−5) is the input, so it goes where the x was. Brackets matter with negative numbers.
   h(−5) = 20(−5)
3. Work it out. Follow BIDMAS.
   = −4

Check. Work it out again on a calculator, typing the brackets exactly as shown: the result is −4. ✓

Watch out. Put brackets round negative inputs. For example (−3)² = 9 but −3² = −9.

Key words. A function is a rule that turns each input into one output. f(x) is read “f of x” and means the output when the input is x.

Method. To find f(a), substitute the number a for x everywhere in the rule, then work out the answer.

1. Write down the function. h(x) is a rule that says what to do to the input x. The letter x is a placeholder for whichever number we put in.
   h(x) = 20x
2. Replace every x with 0.5. The number inside the bracket of h(0.5) is the input, so it goes where the x was.
   h(0.5) = 200.5
3. Work it out. Follow BIDMAS.
   = 40

Check. Work it out again on a calculator, typing the brackets exactly as shown: the result is 40. ✓

Watch out. Put brackets round negative inputs. For example (−3)² = 9 but −3² = −9.

Key words. A function is a rule that turns each input into one output. f(x) is read “f of x” and means the output when the input is x.

Method. To find f(a), substitute the number a for x everywhere in the rule, then work out the answer.

1. Write down the function. h(x) is a rule that says what to do to the input x. The letter x is a placeholder for whichever number we put in.
   h(x) = 20x
2. Replace every x with 0.25. The number inside the bracket of h(0.25) is the input, so it goes where the x was.
   h(0.25) = 200.25
3. Work it out. Follow BIDMAS.
   = 80

Check. Work it out again on a calculator, typing the brackets exactly as shown: the result is 80. ✓

Watch out. Put brackets round negative inputs. For example (−3)² = 9 but −3² = −9.

Key words. A function is a rule that turns each input into one output. f(x) is read “f of x” and means the output when the input is x.

Method. To find f(a), substitute the number a for x everywhere in the rule, then work out the answer.

1. Write down the function. p(x) is a rule that says what to do to the input x. The letter x is a placeholder for whichever number we put in.
   p(x) = 2x² − 5x + 3
2. Replace every x with 1. The number inside the bracket of p(1) is the input, so it goes where the x was.
   p(1) = 2 × 1² − 5 × 1 + 3
3. Work out each part separately (powers first, then multiplying). BIDMAS: indices before multiplication, then add or subtract.
   = 2 − 5 + 3
4. Add and subtract. Work from left to right.
   = 0

Check. Use a different arrangement: x(2x − 5) + 3 with x = 1 gives 1 × (2 × 1 − 5) + 3 = 0. ✓

Watch out. Put brackets round negative inputs. For example (−3)² = 9 but −3² = −9.

Key words. A function is a rule that turns each input into one output. f(x) is read “f of x” and means the output when the input is x.

Method. To find f(a), substitute the number a for x everywhere in the rule, then work out the answer.

1. Write down the function. p(x) is a rule that says what to do to the input x. The letter x is a placeholder for whichever number we put in.
   p(x) = 2x² − 5x + 3
2. Replace every x with −2, using brackets. The number inside the bracket of p(−2) is the input, so it goes where the x was. Brackets matter with negative numbers.
   p(−2) = 2 × (−2)² − 5 × (−2) + 3
3. Work out each part separately (powers first, then multiplying). BIDMAS: indices before multiplication, then add or subtract.
   = 8 + 10 + 3
4. Add and subtract. Work from left to right.
   = 21

Check. Use a different arrangement: x(2x − 5) + 3 with x = −2 gives (−2) × (2 × (−2) − 5) + 3 = 21. ✓

Watch out. Put brackets round negative inputs. For example (−3)² = 9 but −3² = −9.

Key words. A function is a rule that turns each input into one output. f(x) is read “f of x” and means the output when the input is x.

Method. To find f(a), substitute the number a for x everywhere in the rule, then work out the answer.

1. Write down the function. p(x) is a rule that says what to do to the input x. The letter x is a placeholder for whichever number we put in.
   p(x) = 2x² − 5x + 3
2. Replace every x with 3. The number inside the bracket of p(3) is the input, so it goes where the x was.
   p(3) = 2 × 3² − 5 × 3 + 3
3. Work out each part separately (powers first, then multiplying). BIDMAS: indices before multiplication, then add or subtract.
   = 18 − 15 + 3
4. Add and subtract. Work from left to right.
   = 6

Check. Use a different arrangement: x(2x − 5) + 3 with x = 3 gives 3 × (2 × 3 − 5) + 3 = 6. ✓

Watch out. Put brackets round negative inputs. For example (−3)² = 9 but −3² = −9.

Key words. A function is a rule that turns each input into one output. f(x) is read “f of x” and means the output when the input is x.

Method. To find f(a), substitute the number a for x everywhere in the rule, then work out the answer.

1. Write down the function. f(x) is a rule that says what to do to the input x. The letter x is a placeholder for whichever number we put in.
   f(x) = x + 62
2. Replace every x with 8. The number inside the bracket of f(8) is the input, so it goes where the x was.
   f(8) = 8 + 62
3. Work it out. Follow BIDMAS.
   = 7

Check. Work it out again on a calculator, typing the brackets exactly as shown: the result is 7. ✓

Watch out. Put brackets round negative inputs. For example (−3)² = 9 but −3² = −9.

Key words. A function is a rule that turns each input into one output. f(x) is read “f of x” and means the output when the input is x.

Method. To find f(a), substitute the number a for x everywhere in the rule, then work out the answer.

1. Write down the function. f(x) is a rule that says what to do to the input x. The letter x is a placeholder for whichever number we put in.
   f(x) = x + 62
2. Replace every x with −4, using brackets. The number inside the bracket of f(−4) is the input, so it goes where the x was. Brackets matter with negative numbers.
   f(−4) = (−4) + 62
3. Work it out. Follow BIDMAS.
   = 1

Check. Work it out again on a calculator, typing the brackets exactly as shown: the result is 1. ✓

Watch out. Put brackets round negative inputs. For example (−3)² = 9 but −3² = −9.

Key words. A function is a rule that turns each input into one output. f(x) is read “f of x” and means the output when the input is x.

Method. To find f(a), substitute the number a for x everywhere in the rule, then work out the answer.

1. Write down the function. f(x) is a rule that says what to do to the input x. The letter x is a placeholder for whichever number we put in.
   f(x) = x + 62
2. Replace every x with 0. The number inside the bracket of f(0) is the input, so it goes where the x was.
   f(0) = 0 + 62
3. Work it out. Follow BIDMAS.
   = 3

Check. Work it out again on a calculator, typing the brackets exactly as shown: the result is 3. ✓

Watch out. Put brackets round negative inputs. For example (−3)² = 9 but −3² = −9.

Key words. A function is a rule that turns each input into one output. f(x) is read “f of x” and means the output when the input is x.

Method. To find f(a), substitute the number a for x everywhere in the rule, then work out the answer.

1. Write down the function. f(x) is a rule that says what to do to the input x. The letter x is a placeholder for whichever number we put in.
   f(x) = √(x + 9)
2. Replace every x with 16. The number inside the bracket of f(16) is the input, so it goes where the x was.
   f(16) = √(16 + 9)
3. Work it out. Follow BIDMAS.
   = 5

Check. Work it out again on a calculator, typing the brackets exactly as shown: the result is 5. ✓

Watch out. Put brackets round negative inputs. For example (−3)² = 9 but −3² = −9.

Key words. A function is a rule that turns each input into one output. f(x) is read “f of x” and means the output when the input is x.

Method. To find f(a), substitute the number a for x everywhere in the rule, then work out the answer.

1. Write down the function. f(x) is a rule that says what to do to the input x. The letter x is a placeholder for whichever number we put in.
   f(x) = √(x + 9)
2. Replace every x with 0. The number inside the bracket of f(0) is the input, so it goes where the x was.
   f(0) = √(0 + 9)
3. Work it out. Follow BIDMAS.
   = 3

Check. Work it out again on a calculator, typing the brackets exactly as shown: the result is 3. ✓

Watch out. Put brackets round negative inputs. For example (−3)² = 9 but −3² = −9.

Key words. A function is a rule that turns each input into one output. f(x) is read “f of x” and means the output when the input is x.

Method. To find f(a), substitute the number a for x everywhere in the rule, then work out the answer.

1. Write down the function. f(x) is a rule that says what to do to the input x. The letter x is a placeholder for whichever number we put in.
   f(x) = √(x + 9)
2. Replace every x with −5, using brackets. The number inside the bracket of f(−5) is the input, so it goes where the x was. Brackets matter with negative numbers.
   f(−5) = √((−5) + 9)
3. Work it out. Follow BIDMAS.
   = 2

Check. Work it out again on a calculator, typing the brackets exactly as shown: the result is 2. ✓

Watch out. Put brackets round negative inputs. For example (−3)² = 9 but −3² = −9.

Key words. The input is the number that goes into the function and the output is the number that comes out.

Method. To solve f(x) = 17, put the rule in for f(x) and solve the equation for x.

1. Write the equation. We are told that the output is 17, so f(x) = 17. Put the rule in for f(x).
   2x + 5 = 17
2. 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 = 12
3. Divide both sides by 2. The unknown part is being multiplied by 2. The opposite of multiplying by 2 is dividing by 2.
   x = 6
4. Write the answer. x is now on its own.
   x = 6

Check. Put each solution back into the function: f(6) = 2 × 6 + 5 = 17. ✓

Watch out. Do not confuse the two jobs: finding f(3) means put 3 in and get an output; solving f(x) = 3 means the output is 3 and you must find the input.

Key words. The input is the number that goes into the function and the output is the number that comes out.

Method. To solve f(x) = −3, put the rule in for f(x) and solve the equation for x.

1. Write the equation. We are told that the output is −3, so f(x) = −3. Put the rule in for f(x).
   2x + 5 = −3
2. 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 = −8
3. Divide both sides by 2. The unknown part is being multiplied by 2. The opposite of multiplying by 2 is dividing by 2.
   x = −4
4. Write the answer. x is now on its own.
   x = −4

Check. Put each solution back into the function: f(−4) = 2 × (−4) + 5 = −3. ✓

Watch out. Do not confuse the two jobs: finding f(3) means put 3 in and get an output; solving f(x) = 3 means the output is 3 and you must find the input.

Key words. The input is the number that goes into the function and the output is the number that comes out.

Method. To solve f(x) = 0, put the rule in for f(x) and solve the equation for x.

1. Write the equation. We are told that the output is 0, so f(x) = 0. Put the rule in for f(x).
   2x + 5 = 0
2. 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 = −5
3. Divide both sides by 2. The unknown part is being multiplied by 2. The opposite of multiplying by 2 is dividing by 2.
   x = −52
4. Write the answer. x is now on its own.
   x = −2.5

Check. Put each solution back into the function: f(−2.5) = 2 × (−2.5) + 5 = 0. ✓

Watch out. Do not confuse the two jobs: finding f(3) means put 3 in and get an output; solving f(x) = 3 means the output is 3 and you must find the input.

Key words. The input is the number that goes into the function and the output is the number that comes out.

Method. To solve f(x) = 4.5, put the rule in for f(x) and solve the equation for x.

1. Write the equation. We are told that the output is 4.5, so f(x) = 4.5. Put the rule in for f(x).
   2x + 5 = 4.5
2. 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 = −12
3. Divide both sides by 2. The unknown part is being multiplied by 2. The opposite of multiplying by 2 is dividing by 2.
   x = −14
4. Write the answer. x is now on its own.
   x = −0.25

Check. Put each solution back into the function: f(−0.25) = 2 × (−0.25) + 5 = 4.5. ✓

Watch out. Do not confuse the two jobs: finding f(3) means put 3 in and get an output; solving f(x) = 3 means the output is 3 and you must find the input.

Key words. The input is the number that goes into the function and the output is the number that comes out.

Method. To solve g(x) = 18, put the rule in for g(x) and solve the equation for x.

1. Write the equation. We are told that the output is 18, so g(x) = 18. Put the rule in for g(x).
   x² − 7 = 18
2. Add 7 to both sides. The last thing that happens is that 7 is subtracted. The opposite of subtracting 7 is adding 7. Do it to both sides.
   x² = 25
3. Take the square root of both sides. Remember the positive and negative roots. Both 5 and −5 square to give 25.
   x = 5 or x = −5
4. Finish each equation. Solve each of the two equations with inverse operations.
   x = 5
   x = −5

Check. Put each solution back into the function: g(−5) = (−5)² − 7 = 18; g(5) = 5² − 7 = 18. ✓

Watch out. Do not confuse the two jobs: finding f(3) means put 3 in and get an output; solving f(x) = 3 means the output is 3 and you must find the input.

Key words. The input is the number that goes into the function and the output is the number that comes out.

Method. To solve g(x) = 9, put the rule in for g(x) and solve the equation for x.

1. Write the equation. We are told that the output is 9, so g(x) = 9. Put the rule in for g(x).
   x² − 7 = 9
2. Add 7 to both sides. The last thing that happens is that 7 is subtracted. The opposite of subtracting 7 is adding 7. Do it to both sides.
   x² = 16
3. Take the square root of both sides. Remember the positive and negative roots. Both 4 and −4 square to give 16.
   x = 4 or x = −4
4. Finish each equation. Solve each of the two equations with inverse operations.
   x = 4
   x = −4

Check. Put each solution back into the function: g(−4) = (−4)² − 7 = 9; g(4) = 4² − 7 = 9. ✓

Watch out. Do not confuse the two jobs: finding f(3) means put 3 in and get an output; solving f(x) = 3 means the output is 3 and you must find the input.

Key words. The input is the number that goes into the function and the output is the number that comes out.

Method. To solve g(x) = −7, put the rule in for g(x) and solve the equation for x.

1. Write the equation. We are told that the output is −7, so g(x) = −7. Put the rule in for g(x).
   x² − 7 = −7
2. Add 7 to both sides. The last thing that happens is that 7 is subtracted. The opposite of subtracting 7 is adding 7. Do it to both sides.
   x² = 0
3. Take the square root of both sides. Remember the positive and negative roots. Both 0 and −0 square to give 0.
   x = 0 or x = −0
4. Finish each equation. Solve each of the two equations with inverse operations.
   x = 0
   x = 0

Check. Put each solution back into the function: g(0) = 0² − 7 = −7. ✓

Watch out. Do not confuse the two jobs: finding f(3) means put 3 in and get an output; solving f(x) = 3 means the output is 3 and you must find the input.

Key words. The input is the number that goes into the function and the output is the number that comes out.

Method. To solve f(x) = 6, put the rule in for f(x) and solve the equation for x.

1. Write the equation. We are told that the output is 6, so f(x) = 6. Put the rule in for f(x).
   24x = 6
2. 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 6 leaves x on its own.
   24 = 6x
   4 = x
3. Write the answer. x is now on its own.
   x = 4

Check. Put each solution back into the function: f(4) = 244 = 6. ✓

Watch out. Do not confuse the two jobs: finding f(3) means put 3 in and get an output; solving f(x) = 3 means the output is 3 and you must find the input.

Key words. The input is the number that goes into the function and the output is the number that comes out.

Method. To solve f(x) = −3, put the rule in for f(x) and solve the equation for x.

1. Write the equation. We are told that the output is −3, so f(x) = −3. Put the rule in for f(x).
   24x = −3
2. 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 −3 leaves x on its own.
   24 = −3x
   −8 = x
3. Write the answer. x is now on its own.
   x = −8

Check. Put each solution back into the function: f(−8) = 24(−8) = −3. ✓

Watch out. Do not confuse the two jobs: finding f(3) means put 3 in and get an output; solving f(x) = 3 means the output is 3 and you must find the input.

Key words. The input is the number that goes into the function and the output is the number that comes out.

Method. To solve f(x) = 0.5, put the rule in for f(x) and solve the equation for x.

1. Write the equation. We are told that the output is 0.5, so f(x) = 0.5. Put the rule in for f(x).
   24x = 0.5
2. 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 12 leaves x on its own.
   24 = 12x
   48 = x
3. Write the answer. x is now on its own.
   x = 48

Check. Put each solution back into the function: f(48) = 2448 = 0.5. ✓

Watch out. Do not confuse the two jobs: finding f(3) means put 3 in and get an output; solving f(x) = 3 means the output is 3 and you must find the input.

Key words. A function is a rule that turns each input into one output. f(x) is read “f of x” and means the output when the input is x.

Method. To find f(a), substitute the number a for x everywhere in the rule, then work out the answer.

1. Write down the function. C(x) is a rule that says what to do to the input x. The letter x is a placeholder for whichever number we put in.
   C(x) = 1.8x + 2.5
2. Replace every x with 7. The number inside the bracket of C(7) is the input, so it goes where the x was.
   C(7) = 1.8 × 7 + 2.5
3. Work out each part separately (powers first, then multiplying). BIDMAS: indices before multiplication, then add or subtract.
   = 12.6 + 2.5
4. Add and subtract. Work from left to right.
   = 15.1

Check. Work it out again on a calculator, typing the brackets exactly as shown: the result is 15.1. ✓

Watch out. Put brackets round negative inputs. For example (−3)² = 9 but −3² = −9.

Key words. The input is the number that goes into the function and the output is the number that comes out.

Method. To solve C(x) = 24.1, put the rule in for C(x) and solve the equation for x.

1. Write the equation. We are told that the output is 24.1, so C(x) = 24.1. Put the rule in for C(x).
   1.8x + 2.5 = 24.1
2. Subtract 2.5 from both sides. The last thing that happens to the x-part is that 2.5 is added. The opposite of adding 2.5 is subtracting 2.5. Do it to both sides to keep the equation balanced.
   1.8x = 21.6
3. Multiply both sides by 5. The expression is being divided by 5. The opposite of dividing by 5 is multiplying by 5.
   9x = 108
4. Divide both sides by 9. The unknown part is being multiplied by 9. The opposite of multiplying by 9 is dividing by 9.
   x = 12
5. Write the answer. x is now on its own.
   x = 12

Check. Put each solution back into the function: C(12) = 1.8 × 12 + 2.5 = 24.1. ✓

Watch out. Do not confuse the two jobs: finding f(3) means put 3 in and get an output; solving f(x) = 3 means the output is 3 and you must find the input.

Key words. A function is a rule that turns each input into one output. f(x) is read “f of x” and means the output when the input is x.

Method. To find f(a), substitute the number a for x everywhere in the rule, then work out the answer.

1. Write down the function. F(x) is a rule that says what to do to the input x. The letter x is a placeholder for whichever number we put in.
   F(x) = 1.8x + 32
2. Replace every x with 100. The number inside the bracket of F(100) is the input, so it goes where the x was.
   F(100) = 1.8 × 100 + 32
3. Work out each part separately (powers first, then multiplying). BIDMAS: indices before multiplication, then add or subtract.
   = 180 + 32
4. Add and subtract. Work from left to right.
   = 212

Check. Work it out again on a calculator, typing the brackets exactly as shown: the result is 212. ✓

Watch out. Put brackets round negative inputs. For example (−3)² = 9 but −3² = −9.

Key words. A function is a rule that turns each input into one output. f(x) is read “f of x” and means the output when the input is x.

Method. To find f(a), substitute the number a for x everywhere in the rule, then work out the answer.

1. Write down the function. F(x) is a rule that says what to do to the input x. The letter x is a placeholder for whichever number we put in.
   F(x) = 1.8x + 32
2. Replace every x with −10, using brackets. The number inside the bracket of F(−10) is the input, so it goes where the x was. Brackets matter with negative numbers.
   F(−10) = 1.8 × (−10) + 32
3. Work out each part separately (powers first, then multiplying). BIDMAS: indices before multiplication, then add or subtract.
   = −18 + 32
4. Add and subtract. Work from left to right.
   = 14

Check. Work it out again on a calculator, typing the brackets exactly as shown: the result is 14. ✓

Watch out. Put brackets round negative inputs. For example (−3)² = 9 but −3² = −9.

Key words. The input is the number that goes into the function and the output is the number that comes out.

Method. To solve F(x) = 50, put the rule in for F(x) and solve the equation for x.

1. Write the equation. We are told that the output is 50, so F(x) = 50. Put the rule in for F(x).
   1.8x + 32 = 50
2. Subtract 32 from both sides. The last thing that happens to the x-part is that 32 is added. The opposite of adding 32 is subtracting 32. Do it to both sides to keep the equation balanced.
   1.8x = 18
3. Multiply both sides by 5. The expression is being divided by 5. The opposite of dividing by 5 is multiplying by 5.
   9x = 90
4. Divide both sides by 9. The unknown part is being multiplied by 9. The opposite of multiplying by 9 is dividing by 9.
   x = 10
5. Write the answer. x is now on its own.
   x = 10

Check. Put each solution back into the function: F(10) = 1.8 × 10 + 32 = 50. ✓

Watch out. Do not confuse the two jobs: finding f(3) means put 3 in and get an output; solving f(x) = 3 means the output is 3 and you must find the input.

Method. To find g(a + 2), replace x in the rule by a + 2 (in brackets) and simplify.

1. Write the function. g(x) is a rule for the input x. If the input is something else, we put that in the same places.
   g(x) = 3x − 1
2. Replace every x with a + 2. The whole expression in the brackets of g(a + 2) is the input. Use brackets round it.
   g(a + 2) = 3(a + 2) − 1
3. Expand any brackets. Multiply out each bracket, one term at a time. For a squared bracket, multiply the bracket by itself.
   3(a + 2) = 3a + 6
4. Collect like terms. Add together the terms with the same power of the letter.
   g(a + 2) = 3a + 5

Check. Test with a = 2: the input is 2 + 2 = 4, and the original rule gives 11. The answer gives 3 × 2 + 5 = 11. They agree. ✓

Watch out. Replace every x, and expand squares properly: (x + 1)² is (x + 1)(x + 1), not x² + 1.

Method. To find g(2a), replace x in the rule by 2a (in brackets) and simplify.

1. Write the function. g(x) is a rule for the input x. If the input is something else, we put that in the same places.
   g(x) = 3x − 1
2. Replace every x with 2a. The whole expression in the brackets of g(2a) is the input. Use brackets round it.
   g(2a) = 3(2a) − 1
3. Expand any brackets. Multiply out each bracket, one term at a time. For a squared bracket, multiply the bracket by itself.
   3(2a) = 6a
4. Collect like terms. Add together the terms with the same power of the letter.
   g(2a) = 6a − 1

Check. Test with a = 2: the input is 2 × 2 = 4, and the original rule gives 11. The answer gives 6 × 2 − 1 = 11. They agree. ✓

Watch out. Replace every x, and expand squares properly: (x + 1)² is (x + 1)(x + 1), not x² + 1.

Method. To find g(a²), replace x in the rule by a² (in brackets) and simplify.

1. Write the function. g(x) is a rule for the input x. If the input is something else, we put that in the same places.
   g(x) = 3x − 1
2. Replace every x with a². The whole expression in the brackets of g(a²) is the input. Use brackets round it.
   g(a²) = 3(a²) − 1
3. Expand any brackets. Multiply out each bracket, one term at a time. For a squared bracket, multiply the bracket by itself.
   3(a²) = 3a²
4. Collect like terms. Add together the terms with the same power of the letter.
   g(a²) = 3a² − 1

Check. Test with a = 2: the input is 2² = 4, and the original rule gives 11. The answer gives 3 × 2² − 1 = 11. They agree. ✓

Watch out. Replace every x, and expand squares properly: (x + 1)² is (x + 1)(x + 1), not x² + 1.