Inverse and Composite Functions
A composite function combines two functions by substituting the output of one into another, written as fg(x), meaning do g first, then f. An inverse function, written f^-1(x), reverses the effect of f, undoing it to get back to the input; both are tested at GCSE Higher, often alongside algebraic fractions.
Before you start
Make sure you're comfortable with these topics first:
Method
- For a composite function fg(x), work out g(x) first, then substitute that whole expression into f(x) in place of x.
- Read composite function notation carefully: fg(x) means g acts first, not f.
- To find an inverse function f^-1(x), write y = f(x), then rearrange the equation to make x the subject.
- Once rearranged, replace y with x (and x with f^-1(x)) to write the final inverse function.
- Check your inverse is correct by substituting a number into f, then into f^-1, and confirming you get back to the original number.
- For functions written as fractions, clear the fraction by multiplying both sides by the denominator before rearranging.
Worked example
f(x) = 3x - 4 and g(x) = x + 5. Find fg(x) and f^-1(x).
- For fg(x), substitute g(x) = x + 5 into f in place of x: f(x + 5) = 3(x + 5) - 4.
- Expand and simplify: fg(x) = 3x + 15 - 4 = 3x + 11.
- For f^-1(x), write y = 3x - 4 and rearrange to make x the subject: y + 4 = 3x, so x = (y + 4)/3.
- Replace y with x to write the inverse function: f^-1(x) = (x + 4)/3.
- State the final answers: fg(x) = 3x + 11 and f^-1(x) = (x + 4)/3.
Practice questions
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Exam-style questions
Written in the style of a GCSE Maths exam paper, with a full mark scheme.
f(x) = 5x + 4 and g(x) = 2x - 1. Work out fg(3).
f(x) = (x - 7)/2. (a) Find f^-1(x). (b) Work out f^-1(9).
f(x) = x^2 - 1 and g(x) = 2x + 3. (a) Find gf(x). (b) Find fg(x). (c) Work out gf(3) - fg(3).
Free printable worksheet
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