GCSE Maths: Composite functions (Higher)
Part of the GCSE Maths functions and advanced graphs chapter (Higher). Chapter overview · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Composite functions
A composite function is made by joining two functions: the output of the first becomes the input of the second. Suppose g(x) = x + 2 and f(x) = 3x. If we send a number through g and then straight into f, we have one new function.
We write this composite function as fg(x), which means f(g(x)). Be careful about the order. The function nearest the x acts first. So fg(x) means “do g, then do f”. It is read from right to left, which feels backwards at first.
With a number. To find fg(4), first work out g(4) = 4 + 2 = 6, then put the 6 into f: f(6) = 18. So fg(4) = 18.
With algebra. To find fg(x) as one rule, replace the x in f by the whole of g(x), in brackets: f(x) = 3x becomes 3(x + 2) = 3x + 6. So fg(x) = 3x + 6. Check: 3 × 4 + 6 = 18. ✓
The order matters. If we do it the other way, gf(x) = g(f(x)) = 3x + 2, which is a different function. Usually fg and gf are not equal. A function composed with itself is written ff(x) or f2(x).
Solving. To solve an equation such as fg(x) = 20, first build the single function fg(x), then solve it like any other equation.
Inverse of a composite. To undo “do g, then f” you must undo f first and then g. So (fg)−1 = g−1f−1. The order reverses, just like taking off your shoes and then your socks.
Key ideas
- fg(x) means f(g(x)): do g first, then f.
- For a number: work out the inside first, then put that answer into the outside function.
- For algebra: replace the x in the outer function by the whole inner function (in brackets), expand and simplify.
- fg(x) and gf(x) are usually different. ff(x) means f applied twice.
- To solve fg(x) = k: find fg(x) first, then solve the equation.
- (fg)−1 = g−1f−1: the order reverses.
Method. Work from the inside out: put the number into the function nearest it, then put that answer into the other function.
- Remember what fg(3) means. The function nearest the number, g, is applied first. Its output is then used as the input to f.fg(3) = f(g(3))
- Work out the inside: g(3). Put 3 into g.g(3) = 32 = 9
- Use that answer as the input to f: work out f(9). The output 9 goes into f.f(9) = 3 × 9 − 2 = 25
- Write the answer. fg(3) = 25
Check. Another way: find fg(x) = 3x2 − 2 first, then put in 3: 3 × 32 − 2 = 25. ✓
Watch out. fg(3) is not f(3) × g(3). It means f of (g of 3).
Key words. A composite function is two functions joined: the output of the first becomes the input of the second.
Method. fg(x) means: do g first, then f. Replace the x in the outer function by the whole inner function.
- Remember what fg(x) means. fg(x) is the same as f(g(x)). The function written closest to the x (g) is applied first, then f is applied to the result.fg(x) = f(g(x))
- Write down the two functions. g(x) = x2 − 1
f(x) = 2x + 3 - Put g(x) into f: replace the x in f(x) with x2 − 1. g(x) is the output of the first function, and it becomes the input of f. Use brackets.fg(x) = 2(x2 − 1) + 3
- Expand the brackets. Multiply out each bracket, one term at a time.2(x2 − 1) = 2x2 − 2
- Simplify. Collect like terms.fg(x) = 2x2 + 1
Check. Test with x = 2: g(2) = 3, then f(3) = 9. The answer gives 2 × 22 + 1 = 9. They match. ✓
Watch out. Order matters: fg(x) and gf(x) are usually different. The function nearest the x goes first.
Key words. A composite function is two functions joined: the output of the first becomes the input of the second.
Method. gf(x) means: do f first, then g. Replace the x in the outer function by the whole inner function.
- Remember what gf(x) means. gf(x) is the same as g(f(x)). The function written closest to the x (f) is applied first, then g is applied to the result.gf(x) = g(f(x))
- Write down the two functions. f(x) = 2x + 3
g(x) = x2 − 1 - Put f(x) into g: replace the x in g(x) with 2x + 3. f(x) is the output of the first function, and it becomes the input of g. Use brackets.gf(x) = (2x + 3)2 − 1
- Expand the brackets. Multiply out each bracket, one term at a time.(2x + 3)2 = 4x2 + 12x + 9
- Simplify. Collect like terms.gf(x) = 4x2 + 12x + 8
Check. Test with x = 2: f(2) = 7, then g(7) = 48. The answer gives 4 × 22 + 12 × 2 + 8 = 48. They match. ✓
Watch out. Order matters: fg(x) and gf(x) are usually different. The function nearest the x goes first.
Method. First find fg(x) by putting the inner function into the outer one. Then solve fg(x) = 32 as an ordinary equation.
- Remember what fg(x) means. fg(x) is the same as f(g(x)). The function written closest to the x (g) is applied first, then f is applied to the result.fg(x) = f(g(x))
- Write down the two functions. g(x) = x2 + 2
f(x) = 3x − 1 - Put g(x) into f: replace the x in f(x) with x2 + 2. g(x) is the output of the first function, and it becomes the input of f. Use brackets.fg(x) = 3(x2 + 2) − 1
- Expand the brackets. Multiply out each bracket, one term at a time.3(x2 + 2) = 3x2 + 6
- Simplify. Collect like terms.fg(x) = 3x2 + 5
- Write the equation. We are told that the output is 32, so fg(x) = 32. Put the rule in for fg(x).3x2 + 5 = 32
- 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.3x2 = 27
- Divide both sides by 3. The unknown part is being multiplied by 3. The opposite of multiplying by 3 is dividing by 3.x2 = 9
- Take the square root of both sides. Remember the positive and negative roots. Both 3 and −3 square to give 9.x = 3 or x = −3
- Finish each equation. Solve each of the two equations with inverse operations.x = 3
x = −3
Check. Test: fg(−3) = 3 × (−3)2 + 5 = 32; fg(3) = 3 × 32 + 5 = 32. ✓
Watch out. Do not put the number k into the functions one at a time. Build the single function first.
Method. Find fg(x), then invert it. Then find the two separate inverses and compose them in the reverse order.
- Find fg(x). Replace the x in f by g(x) = 3x.fg(x) = 3x + 2
- Invert it: y = 3x + 2. Subtract 2 from both sides, then divide by 3.y − 2 = 3x ⇒ x = y − 23
- Write the inverse. (fg)−1(x) = x − 23
- Find the inverses of f and g separately. Reverse each operation.f−1(x) = x − 2 and g−1(x) = x ÷ 3
- Compose in the reverse order: g−1f−1(x). First undo f (subtract 2), then undo g (divide by 3).g−1(x − 2) = (x − 2) ÷ 3 = x − 23
- Compare. The two answers are the same.(fg)−1 = g−1f−1 ✓
Check. fg(5) = 3 × 5 + 2 = 17, and (fg)−1(17) = (17 − 2) ÷ 3 = 5. ✓
Watch out. If you apply the inverses in the same order as the functions (f−1 first) you get a different, wrong answer.
Watch out
The function nearest the x acts first. fg(2) is not f(2) × g(2) and it is not g(f(2)). It is f(g(2)): work out g(2) first. And when you replace x in the outer function, put brackets round the inner function: 3(x + 2) = 3x + 6, but 3x + 2 is wrong.Exercise 25C Composite functions
Non-calculatorMethod. Work from the inside out: put the number into the function nearest it, then put that answer into the other function.
- Remember what fg(3) means. The function nearest the number, g, is applied first. Its output is then used as the input to f.fg(3) = f(g(3))
- Work out the inside: g(3). Put 3 into g.g(3) = 32 = 9
- Use that answer as the input to f: work out f(9). The output 9 goes into f.f(9) = 2 × 9 + 1 = 19
- Write the answer. fg(3) = 19
Check. Another way: find fg(x) = 2x2 + 1 first, then put in 3: 2 × 32 + 1 = 19. ✓
Watch out. fg(3) is not f(3) × g(3). It means f of (g of 3).
Method. Work from the inside out: put the number into the function nearest it, then put that answer into the other function.
- Remember what gf(3) means. The function nearest the number, f, is applied first. Its output is then used as the input to g.gf(3) = g(f(3))
- Work out the inside: f(3). Put 3 into f.f(3) = 2 × 3 + 1 = 7
- Use that answer as the input to g: work out g(7). The output 7 goes into g.g(7) = 72 = 49
- Write the answer. gf(3) = 49
Check. Another way: find gf(x) = 4x2 + 4x + 1 first, then put in 3: 4 × 32 + 4 × 3 + 1 = 49. ✓
Watch out. gf(3) is not g(3) × f(3). It means g of (f of 3).
Method. Work from the inside out: put the number into the function nearest it, then put that answer into the other function.
- Remember what ff(2) means. The function nearest the number, f, is applied first. Its output is then used as the input to f.ff(2) = f(f(2))
- Work out the inside: f(2). Put 2 into f.f(2) = 2 × 2 + 1 = 5
- Use that answer as the input to f: work out f(5). The output 5 goes into f.f(5) = 2 × 5 + 1 = 11
- Write the answer. ff(2) = 11
Check. Another way: find ff(x) = 4x + 3 first, then put in 2: 4 × 2 + 3 = 11. ✓
Watch out. ff(2) is not f(2) × f(2). It means f of (f of 2).
Method. Work from the inside out: put the number into the function nearest it, then put that answer into the other function.
- Remember what gg(−2) means. The function nearest the number, g, is applied first. Its output is then used as the input to g.gg(−2) = g(g(−2))
- Work out the inside: g(−2). Put −2 into g.g(−2) = (−2)2 = 4
- Use that answer as the input to g: work out g(4). The output 4 goes into g.g(4) = 42 = 16
- Write the answer. gg(−2) = 16
Check. Another way: find gg(x) = x4 first, then put in −2: (−2)4 = 16. ✓
Watch out. gg(−2) is not g(−2) × g(−2). It means g of (g of −2).
Key words. A composite function is two functions joined: the output of the first becomes the input of the second.
Method. fg(x) means: do g first, then f. Replace the x in the outer function by the whole inner function.
- Remember what fg(x) means. fg(x) is the same as f(g(x)). The function written closest to the x (g) is applied first, then f is applied to the result.fg(x) = f(g(x))
- Write down the two functions. g(x) = 3x
f(x) = x − 4 - Put g(x) into f: replace the x in f(x) with 3x. g(x) is the output of the first function, and it becomes the input of f. Use brackets.fg(x) = 3x − 4
- Simplify. Write the answer in its simplest form.fg(x) = 3x − 4
Check. Test with x = 2: g(2) = 6, then f(6) = 2. The answer gives 3 × 2 − 4 = 2. They match. ✓
Watch out. Order matters: fg(x) and gf(x) are usually different. The function nearest the x goes first.
Key words. A composite function is two functions joined: the output of the first becomes the input of the second.
Method. gf(x) means: do f first, then g. Replace the x in the outer function by the whole inner function.
- Remember what gf(x) means. gf(x) is the same as g(f(x)). The function written closest to the x (f) is applied first, then g is applied to the result.gf(x) = g(f(x))
- Write down the two functions. f(x) = x − 4
g(x) = 3x - Put f(x) into g: replace the x in g(x) with x − 4. f(x) is the output of the first function, and it becomes the input of g. Use brackets.gf(x) = 3(x − 4)
- Expand the brackets. Multiply out each bracket, one term at a time.3(x − 4) = 3x − 12
- Simplify. Collect like terms.gf(x) = 3x − 12
Check. Test with x = 2: f(2) = −2, then g(−2) = −6. The answer gives 3 × 2 − 12 = −6. They match. ✓
Watch out. Order matters: fg(x) and gf(x) are usually different. The function nearest the x goes first.
Key words. A composite function is two functions joined: the output of the first becomes the input of the second.
Method. fg(x) means: do g first, then f. Replace the x in the outer function by the whole inner function.
- Remember what fg(x) means. fg(x) is the same as f(g(x)). The function written closest to the x (g) is applied first, then f is applied to the result.fg(x) = f(g(x))
- Write down the two functions. g(x) = x2 − 1
f(x) = x + 5 - Put g(x) into f: replace the x in f(x) with x2 − 1. g(x) is the output of the first function, and it becomes the input of f. Use brackets.fg(x) = x2 − 1 + 5
- Simplify. Write the answer in its simplest form.fg(x) = x2 + 4
Check. Test with x = 2: g(2) = 3, then f(3) = 8. The answer gives 22 + 4 = 8. They match. ✓
Watch out. Order matters: fg(x) and gf(x) are usually different. The function nearest the x goes first.
Key words. A composite function is two functions joined: the output of the first becomes the input of the second.
Method. gf(x) means: do f first, then g. Replace the x in the outer function by the whole inner function.
- Remember what gf(x) means. gf(x) is the same as g(f(x)). The function written closest to the x (f) is applied first, then g is applied to the result.gf(x) = g(f(x))
- Write down the two functions. f(x) = x + 5
g(x) = x2 − 1 - Put f(x) into g: replace the x in g(x) with x + 5. f(x) is the output of the first function, and it becomes the input of g. Use brackets.gf(x) = (x + 5)2 − 1
- Expand the brackets. Multiply out each bracket, one term at a time.(x + 5)2 = x2 + 10x + 25
- Simplify. Collect like terms.gf(x) = x2 + 10x + 24
Check. Test with x = 2: f(2) = 7, then g(7) = 48. The answer gives 22 + 10 × 2 + 24 = 48. They match. ✓
Watch out. Order matters: fg(x) and gf(x) are usually different. The function nearest the x goes first.
Key words. A composite function is two functions joined: the output of the first becomes the input of the second.
Method. fg(x) means: do g first, then f. Replace the x in the outer function by the whole inner function.
- Remember what fg(x) means. fg(x) is the same as f(g(x)). The function written closest to the x (g) is applied first, then f is applied to the result.fg(x) = f(g(x))
- Write down the two functions. g(x) = x2
f(x) = 3x − 2 - Put g(x) into f: replace the x in f(x) with x2. g(x) is the output of the first function, and it becomes the input of f. Use brackets.fg(x) = 3(x2) − 2
- Expand the brackets. Multiply out each bracket, one term at a time.3(x2) = 3x2
- Simplify. Collect like terms.fg(x) = 3x2 − 2
Check. Test with x = 2: g(2) = 1, then f(1) = 1. The answer gives 3 × 22 − 2 = 1. They match. ✓
Watch out. Order matters: fg(x) and gf(x) are usually different. The function nearest the x goes first.
Key words. A composite function is two functions joined: the output of the first becomes the input of the second.
Method. gf(x) means: do f first, then g. Replace the x in the outer function by the whole inner function.
- Remember what gf(x) means. gf(x) is the same as g(f(x)). The function written closest to the x (f) is applied first, then g is applied to the result.gf(x) = g(f(x))
- Write down the two functions. f(x) = 3x − 2
g(x) = x2 - Put f(x) into g: replace the x in g(x) with 3x − 2. f(x) is the output of the first function, and it becomes the input of g. Use brackets.gf(x) = 3x − 22
- Expand the brackets. Multiply out each bracket, one term at a time.3x − 22 = 3x2 − 1
- Simplify. Collect like terms.gf(x) = 3x2 − 1
Check. Test with x = 2: f(2) = 4, then g(4) = 2. The answer gives 3 × 22 − 1 = 2. They match. ✓
Watch out. Order matters: fg(x) and gf(x) are usually different. The function nearest the x goes first.
Key words. A composite function is two functions joined: the output of the first becomes the input of the second.
Method. ff(x) means: do f first, then f. Replace the x in the outer function by the whole inner function.
- Remember what ff(x) means. ff(x) is the same as f(f(x)). The function written closest to the x (f) is applied first, then f is applied to the result.ff(x) = f(f(x))
- Write down the two functions. f(x) = 2x + 3
f(x) = 2x + 3 - Put f(x) into f: replace the x in f(x) with 2x + 3. f(x) is the output of the first function, and it becomes the input of f. Use brackets.ff(x) = 2(2x + 3) + 3
- Expand the brackets. Multiply out each bracket, one term at a time.2(2x + 3) = 4x + 6
- Simplify. Collect like terms.ff(x) = 4x + 9
Check. Test with x = 2: f(2) = 7, then f(7) = 17. The answer gives 4 × 2 + 9 = 17. They match. ✓
Watch out. Order matters: ff(x) and ff(x) are usually different. The function nearest the x goes first.
Key words. A composite function is two functions joined: the output of the first becomes the input of the second.
Method. gg(x) means: do g first, then g. Replace the x in the outer function by the whole inner function.
- Remember what gg(x) means. gg(x) is the same as g(g(x)). The function written closest to the x (g) is applied first, then g is applied to the result.gg(x) = g(g(x))
- Write down the two functions. g(x) = x2
g(x) = x2 - Put g(x) into g: replace the x in g(x) with x2. g(x) is the output of the first function, and it becomes the input of g. Use brackets.gg(x) = (x2)2
- Expand the brackets. Multiply out each bracket, one term at a time.(x2)2 = x4
- Simplify. Collect like terms.gg(x) = x4
Check. Test with x = 2: g(2) = 4, then g(4) = 16. The answer gives 24 = 16. They match. ✓
Watch out. Order matters: gg(x) and gg(x) are usually different. The function nearest the x goes first.
Key words. A composite function is two functions joined: the output of the first becomes the input of the second.
Method. hh(x) means: do h first, then h. Replace the x in the outer function by the whole inner function.
- Remember what hh(x) means. hh(x) is the same as h(h(x)). The function written closest to the x (h) is applied first, then h is applied to the result.hh(x) = h(h(x))
- Write down the two functions. h(x) = −2x + 1
h(x) = −2x + 1 - Put h(x) into h: replace the x in h(x) with −2x + 1. h(x) is the output of the first function, and it becomes the input of h. Use brackets.hh(x) = −2(−2x + 1) + 1
- Expand the brackets. Multiply out each bracket, one term at a time.−2(−2x + 1) = 4x − 2
- Simplify. Collect like terms.hh(x) = 4x − 1
Check. Test with x = 2: h(2) = −3, then h(−3) = 7. The answer gives 4 × 2 − 1 = 7. They match. ✓
Watch out. Order matters: hh(x) and hh(x) are usually different. The function nearest the x goes first.
Method. First find fg(x) by putting the inner function into the outer one. Then solve fg(x) = 9 as an ordinary equation.
- Remember what fg(x) means. fg(x) is the same as f(g(x)). The function written closest to the x (g) is applied first, then f is applied to the result.fg(x) = f(g(x))
- Write down the two functions. g(x) = x − 3
f(x) = 2x + 1 - Put g(x) into f: replace the x in f(x) with x − 3. g(x) is the output of the first function, and it becomes the input of f. Use brackets.fg(x) = 2(x − 3) + 1
- Expand the brackets. Multiply out each bracket, one term at a time.2(x − 3) = 2x − 6
- Simplify. Collect like terms.fg(x) = 2x − 5
- Write the equation. We are told that the output is 9, so fg(x) = 9. Put the rule in for fg(x).2x − 5 = 9
- Add 5 to both sides. The last thing that happens is that 5 is subtracted. The opposite of subtracting 5 is adding 5. Do it to both sides.2x = 14
- Divide both sides by 2. The unknown part is being multiplied by 2. The opposite of multiplying by 2 is dividing by 2.x = 7
- Write the answer. x is now on its own.x = 7
Check. Test: fg(7) = 2 × 7 − 5 = 9. ✓
Watch out. Do not put the number k into the functions one at a time. Build the single function first.
Method. First find gf(x) by putting the inner function into the outer one. Then solve gf(x) = 15 as an ordinary equation.
- Remember what gf(x) means. gf(x) is the same as g(f(x)). The function written closest to the x (f) is applied first, then g is applied to the result.gf(x) = g(f(x))
- Write down the two functions. f(x) = x + 1
g(x) = x2 − 1 - Put f(x) into g: replace the x in g(x) with x + 1. f(x) is the output of the first function, and it becomes the input of g. Use brackets.gf(x) = (x + 1)2 − 1
- Expand the brackets. Multiply out each bracket, one term at a time.(x + 1)2 = x2 + 2x + 1
- Simplify. Collect like terms.gf(x) = x2 + 2x
- Write the equation. We are told that the output is 15, so gf(x) = 15. Put the rule in for gf(x).x2 + 2x = 15
- Move everything to one side so that the other side is 0. To factorise, a quadratic equation must equal 0. Subtract 15 from both sides.x2 + 2x − 15 = 0
- Factorise the quadratic. If two things multiply to give 0, at least one of them must be 0. Factorising turns the equation into a product that equals 0.(x − 3)(x + 5) = 0
- Set each bracket equal to 0 and solve. One bracket must be 0.x − 3 = 0 ⇒ x = 3
x + 5 = 0 ⇒ x = −5 - Write the solutions. A quadratic equation can have two solutions, so give both.x = −5 or x = 3
Check. Test: gf(−5) = (−5)2 + 2 × (−5) = 15; gf(3) = 32 + 2 × 3 = 15. ✓
Watch out. Do not put the number k into the functions one at a time. Build the single function first.
Method. Work out both composite functions in each case and compare them.
- (a) Work out fg(x). Replace x in f by x − 5.fg(x) = (x − 5) + 3 = x − 2
- (a) Work out gf(x). Replace x in g by x + 3.gf(x) = (x + 3) − 5 = x − 2
- (a) Compare. Both equal x − 2, so fg(x) = gf(x). ✓
- (b) Work out pq(x). Replace x in p by x + 1.pq(x) = 2(x + 1) = 2x + 2
- (b) Work out qp(x). Replace x in q by 2x.qp(x) = 2x + 1 = 2x + 1
- (b) Compare. 2x + 2 and 2x + 1 are different (for example when x = 0 they give 2 and 1).pq(x) ≠ qp(x). ✓
Check. In (a) both operations are just adding or subtracting, so the order does not matter. In (b) one multiplies and one adds, so the order does matter. ✓
Watch out. Two functions that both only add or subtract can be done in either order. Multiplying and adding together cannot.
Method. Do the reduction first (f), then add the tax (g), which is gf(x). Use the multiplier form.
- Find gf(x). f acts first, so put 0.8x into g.gf(x) = 1.2 × (0.8x) = 0.96x
- Interpret the multiplier. Multiplying by 0.96 is the same as keeping 96% of the price.final price = 96% of the original price
- So the final price is not the same as the original. A 20% decrease followed by a 20% increase does not cancel, because the 20% increase is taken of a smaller amount.It is 4% less.
- Work out the price of the £50 item. Put 50 in.gf(50) = 0.96 × 50 = £48
Check. Step by step: £50 × 0.8 = £40, then £40 × 1.2 = £48. ✓
Watch out. Percentage increases and decreases do not cancel out. Use multipliers.
Key words. A composite function is two functions joined: the output of the first becomes the input of the second.
Method. fg(x) means: do g first, then f. Replace the x in the outer function by the whole inner function.
- Remember what fg(x) means. fg(x) is the same as f(g(x)). The function written closest to the x (g) is applied first, then f is applied to the result.fg(x) = f(g(x))
- Write down the two functions. g(x) = x + 2
f(x) = 1x - Put g(x) into f: replace the x in f(x) with x + 2. g(x) is the output of the first function, and it becomes the input of f. Use brackets.fg(x) = 1x + 2
- Simplify. Write the answer in its simplest form.fg(x) = 1x + 2
Check. Test with x = 2: g(2) = 4, then f(4) = 0.25. The answer gives 12 + 2 = 0.25. They match. ✓
Watch out. Order matters: fg(x) and gf(x) are usually different. The function nearest the x goes first.
Key words. A composite function is two functions joined: the output of the first becomes the input of the second.
Method. gf(x) means: do f first, then g. Replace the x in the outer function by the whole inner function.
- Remember what gf(x) means. gf(x) is the same as g(f(x)). The function written closest to the x (f) is applied first, then g is applied to the result.gf(x) = g(f(x))
- Write down the two functions. f(x) = 1x
g(x) = x + 2 - Put f(x) into g: replace the x in g(x) with 1x. f(x) is the output of the first function, and it becomes the input of g. Use brackets.gf(x) = 1x + 2
- Simplify. Write the answer in its simplest form.gf(x) = 1x + 2
Check. Test with x = 2: f(2) = 0.5, then g(0.5) = 2.5. The answer gives 12 + 2 = 2.5. They match. ✓
Watch out. Order matters: fg(x) and gf(x) are usually different. The function nearest the x goes first.
Method. Work from the inside out: put the number into the function nearest it, then put that answer into the other function.
- Remember what fg(13) means. The function nearest the number, g, is applied first. Its output is then used as the input to f.fg(13) = f(g(13))
- Work out the inside: g(13). Put 13 into g.g(13) = 13 − 4 = 9
- Use that answer as the input to f: work out f(9). The output 9 goes into f.f(9) = √9 = 3
- Write the answer. fg(13) = 3
Check. Another way: find fg(x) = √(x − 4) first, then put in 13: √(13 − 4) = 3. ✓
Watch out. fg(13) is not f(13) × g(13). It means f of (g of 13).
Method. Work from the inside out: put the number into the function nearest it, then put that answer into the other function.
- Remember what fg(20) means. The function nearest the number, g, is applied first. Its output is then used as the input to f.fg(20) = f(g(20))
- Work out the inside: g(20). Put 20 into g.g(20) = 20 − 4 = 16
- Use that answer as the input to f: work out f(16). The output 16 goes into f.f(16) = √16 = 4
- Write the answer. fg(20) = 4
Check. Another way: find fg(x) = √(x − 4) first, then put in 20: √(20 − 4) = 4. ✓
Watch out. fg(20) is not f(20) × g(20). It means f of (g of 20).
Key words. A composite function is two functions joined: the output of the first becomes the input of the second.
Method. fg(x) means: do g first, then f. Replace the x in the outer function by the whole inner function.
- Remember what fg(x) means. fg(x) is the same as f(g(x)). The function written closest to the x (g) is applied first, then f is applied to the result.fg(x) = f(g(x))
- Write down the two functions. g(x) = x − 4
f(x) = √x - Put g(x) into f: replace the x in f(x) with x − 4. g(x) is the output of the first function, and it becomes the input of f. Use brackets.fg(x) = √(x − 4)
- Simplify. Write the answer in its simplest form.fg(x) = √(x − 4)
Check. Test with x = 5: g(5) = 1, then f(1) = 1. The answer gives √(5 − 4) = 1. They match. ✓
Watch out. Order matters: fg(x) and gf(x) are usually different. The function nearest the x goes first.
Method. fg(x) = f(g(x)) = 2g(x) + 1. Set this equal to 6x + 7 and solve for g(x).
- Write fg(x) using the rule for f. Replace x in 2x + 1 by g(x).fg(x) = 2g(x) + 1
- Set it equal to the given expression. 2g(x) + 1 = 6x + 7
- Subtract 1 from both sides. 2g(x) = 6x + 6
- Divide both sides by 2. Divide every term.g(x) = 3x + 3
Check. f(g(x)) = 2(3x + 3) + 1 = 6x + 7. ✓
Watch out. Treat g(x) like a single letter, then rearrange.
Method. Work out both and compare. The correct meaning is: do g first, then f.
- Work out fg(2) correctly. Do g first.g(2) = 22 = 4, then f(4) = 4 + 3 = 7
- Work out Aisha’s version. f(2) = 5 and g(2) = 4.5 × 4 = 20
- Compare. 7 and 20 are different.So Aisha is wrong: fg(2) = 7, not 20.
Check. fg(x) = x2 + 3, and 22 + 3 = 7. ✓
Watch out. fg(x) is two functions joined, not two functions multiplied.
Method. Work out g(1) in terms of k, put that into f, and form an equation.
- Find g(1). Put 1 in place of x.g(1) = 12 + k = 1 + k
- Find fg(1) = f(1 + k). Put the output of g into f, with brackets.f(1 + k) = 3(1 + k) + 2 = 5 + 3k
- Set it equal to 20. 5 + 3k = 20
- Solve. Subtract 5, then divide by 3.3k = 15, so k = 5
Check. With k = 5: g(1) = 6, f(6) = 20. ✓
Watch out. Use brackets when you put 1 + k into f.
Method. Find fg(x) and gf(x) in terms of a, then set them equal.
- Find fg(x). Replace x in f by 3x + 4.fg(x) = 2(3x + 4) + a = a + 6x + 8
- Find gf(x). Replace x in g by 2x + a.gf(x) = 3(2x + a) + 4 = 3a + 6x + 4
- Set them equal. The 6x terms match, so compare the number terms.8 + a = 3a + 4
- Solve. Subtract a and 4 from both sides: 4 = 2a.a = 2
Check. With a = 2: fg(x) = 6x + 10 and gf(x) = 6x + 10. ✓
Watch out. Both sides contain 6x, so only the numbers decide the value of a.
Method. Apply the functions one at a time, starting with the one nearest the x, and simplify after each.
- Remember the order. In hgf(x) the function written nearest the x acts first. Read it from the right: f, then g, then h.hgf(x) = h(g(f(x)))
- Apply f. Replace the x in f(x) = x + 1 by the current expression (x).f(x) gives x + 1
- Apply g. Replace the x in g(x) = 2x by the current expression (x + 1).g(x) gives 2x + 2
- Apply h. Replace the x in h(x) = x2 by the current expression (2x + 2).h(x) gives 4x2 + 8x + 4
- Write the answer. hgf(x) = 4x2 + 8x + 4
Check. Test with x = 1: apply the functions in turn to get 16; the answer gives 4 × 12 + 8 × 1 + 4 = 16. ✓
Watch out. Work from right to left in the name: hgf means f first, then g, then h.