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Composite and Inverse Functions

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AQA Level 2 Further Maths practice questions, each with a hint and a full worked solution.

Have a pen, paper and a calculator handy, to work out your answers.

Question 11 mark

\mathrm{f}(x)=2x-5 and \mathrm{g}(x)=x^2+1

Which expression is \mathrm{gf}(x)?

Select the correct answer.

Hint

\mathrm{gf}(x) means apply f first, then put the whole of \mathrm{f}(x) into g.

Worked solution
  1. \mathrm{gf}(x)=\mathrm{g}(2x-5), so do f first.
  2. \mathrm{g}(2x-5)=(2x-5)^2+1
  3. (2x-5)^2=4x^2-20x+25
  4. \mathrm{gf}(x)=4x^2-20x+26
  5. (2x^2-3 is \mathrm{fg}(x), the wrong order; 4x^2+26 and 4x^2-10x+26 come from squaring the bracket wrongly.)

Question 23 marks

\mathrm{f}(x)=\dfrac{3x}{x+2} x\neq -2

Work out \mathrm{f}^{-1}(x)

Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

Hint

Write y=\dfrac{3x}{x+2}, multiply both sides by (x+2), then collect the x terms on one side and factorise.

Worked solution
  1. Let y=\dfrac{3x}{x+2}
  2. Multiply by (x+2): \;y(x+2)=3x
  3. Expand: \;xy+2y=3x
  4. Collect the x terms: \;2y=3x-xy
  5. Factorise: \;2y=x(3-y)
  6. Divide: \;x=\dfrac{2y}{3-y}
  7. Swap back to x: \;\mathrm{f}^{-1}(x)=\dfrac{2x}{3-x}

Question 3Challenge6 marks

\mathrm{f}(x)=2x+k, where k is a constant.

\mathrm{g}(x)=x^2+4

\mathrm{f}^{-1}(15)=4

(a)

Work out the value of k.

2 marks

(b)

Solve \;\mathrm{fg}(x)=\mathrm{gf}(x)

Give your answers to 2 decimal places.

4 marks

Give every value, separated by commas

Hint

\mathrm{f}^{-1}(15)=4 tells you that \mathrm{f}(4)=15. For (b), do g first in \mathrm{fg}(x) and f first in \mathrm{gf}(x), and expand (2x+7)^2 carefully.

Worked solution

Part (a)

  1. \mathrm{f}^{-1}(15)=4 means \mathrm{f}(4)=15
  2. 2(4)+k=15
  3. 8+k=15
  4. k=7

Part (b)

  1. With k=7: \;\mathrm{f}(x)=2x+7
  2. \mathrm{fg}(x)=2(x^2+4)+7=2x^2+15
  3. \mathrm{gf}(x)=(2x+7)^2+4=4x^2+28x+53
  4. Set equal: \;2x^2+15=4x^2+28x+53
  5. Rearrange: \;2x^2+28x+38=0, so x^2+14x+19=0
  6. Quadratic formula: \;x=\dfrac{-14\pm\sqrt{14^2-4(1)(19)}}{2}=\dfrac{-14\pm\sqrt{120}}{2}
  7. x=-1.52 or x=-12.48 (2 d.p.)

More on this topic: Composite and Inverse Functions worksheet with full solutions

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