Composite functions
A composite function applies one function and then another. The notation fg(x) means “do g first, then f” — that is, f(g(x)). The order matters: fg and gf are usually different.
Key idea
fg(x) = f(g(x)) — work from the inside out. Substitute the whole of g(x) into f.
Worked example
Let f(x) = 2x + 1 and g(x) = x². Then fg(x) = f(x²) = 2x² + 1, while gf(x) = g(2x + 1) = (2x + 1)². These are not the same, showing order matters.
Inverse functions
The inverse f⁻¹ reverses f: if f sends a to b, then f⁻¹ sends b back to a. An inverse exists only when f is one-one. To find f⁻¹, write y = f(x), swap the roles of x and y, then make y the subject.
Worked example
For f(x) = 2x + 1: let y = 2x + 1, so x = (y − 1)⁄2. Hence f⁻¹(x) = (x − 1)⁄2. Check: f(f⁻¹(x)) = 2·(x−1)⁄2 + 1 = x.
Remember
- fg(x) = f(g(x)); apply the right-hand function first.
- An inverse exists only for a one-one function.
- The graph of f⁻¹ is the reflection of f in the line y = x.
- f(f⁻¹(x)) = x and f⁻¹(f(x)) = x.