Functions: Composite and Inverse Functions
A composite function fg(x) means applying g first and then applying f to the result, so fg(x) = f(g(x)); the order matters, since gf(x) is usually different. An inverse function f^-1(x) reverses the effect of f(x), and is found by writing y = f(x), rearranging to make x the subject, then swapping x and y to give f^-1(x) in terms of x. Both ideas are core to AQA Level 2 Further Maths.
Before you start
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Method
- To find a composite function fg(x), first work out g(x), then substitute that whole expression into f(x) in place of x.
- Work carefully with brackets when substituting, expanding any powers or products that result.
- To find an inverse function, start by writing y = f(x).
- Rearrange the equation to make x the subject, undoing the operations in reverse order.
- Swap x and y (or simply rewrite in terms of x) to state f^-1(x).
- Check your inverse by picking a value a, working out f(a), then confirming that f^-1 of that result returns you to a.
Worked example
f(x) = 4x - 3 and g(x) = x + 5. Find fg(x), and hence find fg(2).
- Substitute g(x) into f(x): fg(x) = f(x + 5) = 4(x + 5) - 3.
- Expand the bracket: 4x + 20 - 3.
- Simplify: fg(x) = 4x + 17.
- Substitute x = 2: fg(2) = 4(2) + 17.
- Final answer: fg(2) = 25.
Practice questions
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Q1f(x) = 3x + 1 and g(x) = x - 2. Find f(5).Show answer
Answer: 16
Q2f(x) = 3x + 1 and g(x) = x - 2. Find g(10).Show answer
Answer: 8
Q3f(x) = 3x + 1 and g(x) = x - 2. Find fg(4).Show answer
Answer: 7 (g(4) = 2, then f(2) = 7)
Q4f(x) = 2x - 5. Find f^-1(x).Show answer
Answer: f^-1(x) = (x + 5)/2
Q5f(x) = (x - 3)/4. Find f^-1(x) and evaluate f^-1(5).Show answer
Answer: f^-1(x) = 4x + 3; f^-1(5) = 23
Q6f(x) = 4x - 1 and g(x) = x^2 + 2. Find gf(x), giving your answer in the form ax^2 + bx + c.Show answer
Answer: gf(x) = 16x^2 - 8x + 3
Exam-style questions
Written in the style of a GCSE Further Maths exam paper, with a full mark scheme.
f(x) = 2x + 7 and g(x) = x^2 - 3. Find fg(x) and gf(x), giving each answer in its simplest form.
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f(x) = (2x - 3)/5. Find f^-1(x), and hence evaluate f^-1(1).
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f(x) = 2x - 3 and g(x) = x^2 + k, where k is a constant. (a) Find fg(x) in terms of x and k. (b) Given that fg(4) = 15, find the value of k. (c) Hence find gf(x), giving your answer in the form ax^2 + bx + c.
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Free printable worksheet
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