Functions: Composite and Inverse Functions
A function is a rule that assigns exactly one output to each input, written f(x). A composite function applies two functions in succession: fg(x) means apply g first and then f, which is the opposite order to the way it is read, and that is the single most common error in the topic. An inverse function, written f with a superscript -1, reverses the effect of f, so f applied to its own inverse returns the original input. An inverse exists only where each output comes from exactly one input. Finding an inverse is a mechanical process: write y = f(x), swap x and y, then rearrange to make y the subject.
Before you start
No specific prerequisites - this is a good place to start.
Method
- Read composite notation right to left: in fg(x) the function nearest to x acts first. Writing fg(x) = f(g(x)) explicitly before substituting prevents the order error.
- To evaluate a composite at a number, work inside out: find the inner function's value first, then apply the outer function to that number.
- To find a composite as an expression, substitute the whole inner expression into every x of the outer function, using brackets so that squaring or multiplying applies to all of it.
- To find an inverse, set y = f(x), swap every x and y, then rearrange to make y the subject. Write the answer in terms of x.
- Check an inverse by composing it with the original: f of the inverse of x should simplify to x. If it does not, the rearrangement went wrong.
- State any restriction the inverse needs. A square root gives a non-negative output, and a denominator cannot be zero, so those values are excluded from the domain.
Worked example
f(x) = 2x + 3 and g(x) = x^2. Find fg(x), gf(x), and the inverse of f.
- For fg(x), g acts first: fg(x) = f(g(x)) = f(x^2).
- Substitute x^2 into f: f(x^2) = 2(x^2) + 3 = 2x^2 + 3.
- For gf(x), f acts first: gf(x) = g(f(x)) = g(2x + 3) = (2x + 3)^2, which expands to 4x^2 + 12x + 9. Note that fg and gf are different, which is normally the case.
- For the inverse, write y = 2x + 3, then swap the letters to get x = 2y + 3.
- Rearrange for y: x - 3 = 2y, so y = (x - 3)/2, giving the inverse of f as (x - 3)/2.
- Check: f applied to (x - 3)/2 gives 2 times (x - 3)/2 plus 3 = (x - 3) + 3 = x, so the inverse is correct.
Practice questions
Try each question, then tap to reveal the answer.
Q1f(x) = 3x - 1. Find f(4).Show answer
Answer: 3(4) - 1 = 11.
Q2f(x) = x + 5 and g(x) = 2x. Find fg(3).Show answer
Answer: g acts first: g(3) = 6, then f(6) = 11.
Q3Which function acts first in gf(x)?Show answer
Answer: f, because it is the one written next to the x.
Q4f(x) = 4x + 2. Find the inverse of f.Show answer
Answer: y = 4x + 2, swap to x = 4y + 2, so y = (x - 2)/4.
Q5f(x) = x^3. Find the inverse of f.Show answer
Answer: The cube root of x.
Q6f(x) = 2x and g(x) = x + 1. Show that fg(x) and gf(x) are different.Show answer
Answer: fg(x) = f(x + 1) = 2(x + 1) = 2x + 2, while gf(x) = g(2x) = 2x + 1. They differ by 1 for every x.
Q7How can you check an inverse function is correct?Show answer
Answer: Compose it with the original function. If f applied to the inverse of x simplifies to x, the inverse is right.
Exam-style questions
Written in the style of a IGCSE Maths exam paper, with a full mark scheme.
f(x) = 5x - 4 and g(x) = x^2 + 1. (a) Find gf(2). (b) Find an expression for fg(x), simplifying your answer. (c) Find the inverse of f.
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f(x) = 1/(x - 2). Explain why x = 2 must be excluded from the domain of f, and find the inverse of f, stating the value that must be excluded from ITS domain.
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Free printable worksheet
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