Worked example 1
Find a composite
f(x) = 2x + 3 and g(x) = x² − 1. Find f(g(x))
- Start with g(x) = x² − 1
- Put this into f: f(g(x)) = 2(x² − 1) + 3
- Simplify.
Answer: f(g(x)) = 2x² + 1
Back to Higher Mathematics
Higher Mathematics
Composite and inverse functions
Build f(g(x)), reverse simple functions and handle domain restrictions for inverses.
Before you start
- Rearrange linear equations confidently.
- Know the difference between f(g(x)) and g(f(x)).
- Use function notation with brackets.
Explanation
A composite function uses one function as the input of another. For f(g(x)), apply g first, then f.
An inverse function reverses the mapping. For a one-to-one function, swap x and y, then rearrange for y.
Visual support
Method and rules
- f(g(x)) means apply g, then f.
- For an inverse: write y = f(x), swap x and y, rearrange for y.
- Check: f(f⁻¹(x)) = x
Worked examples
Worked example 2
Find an inverse
Find f⁻¹(x) for f(x) = 3x − 5.
- Write y = 3x − 5
- Swap x and y: x = 3y − 5
- Rearrange for y.
So: f⁻¹(x) = x + 53
Worked example 3
Check an inverse
Check the inverse of f(x) = 3x − 5
- Calculate fx + 53
- Simplify 3x + 53 − 5.
Answer: The result is x, so the inverse is correct.
Watch out
- Applying the functions in the wrong order.
- Forgetting to swap x and y before rearranging.
- Calling an inverse valid without considering whether the original function is one-to-one on the stated domain.
Exam reminder
Higher exam questions usually reward method setup before the final value. State the rule you are using, keep exact values where possible and check that the answer matches the question wording.