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
Higher Mathematics
Higher Maths Composite and inverse functions
Practise composite and inverse functions for Scottish Higher Mathematics with worked examples, clear methods and original interactive questions. Build composite functions, find inverses and state suitable domains and ranges.
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.
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) =
Worked example 3
Check an inverse
Check the inverse of f(x) = 3x − 5
- Calculate f
- Simplify 3 − 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.
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