Functions
Functions, domain and range
- A function $f(x)$ sends each input to exactly one output.
- the domain = allowed inputs; the range = the outputs produced.
- A function is one-one if different inputs always give different outputs — only then does it have an inverse $f^{-1}$.
- Composition $fg(x) = f(g(x))$ — do $g$ first, then $f$.
Practice
If f(x) = x² and g(x) = x + 1, what is fg(2) = f(g(2))?
g(2) = 3, then f(3) = 3² = 9.
Practice
Only a one-one function has an inverse function.
If two inputs gave the same output, the reverse rule would be ambiguous — so only one-one functions are invertible.
Finding an inverse
- To find $f^{-1}$: write $y = f(x)$, make $x$ the subject, then swap letters.
- Example: $f(x) = 2x + 6$ → $x = \dfrac{y-6}{2}$ → $f^{-1}(x) = \dfrac{x-6}{2}$.
- The graph of $y = f^{-1}(x)$ is the reflection of $y = f(x)$ in the line $y = x$.
Practice
For f(x) = 2x + 6, the inverse is f⁻¹(x) = (x − 6)/2. What is f⁻¹(10)?
f⁻¹(10) = (10 − 6)/2 = 4/2 = 2.
Practice
The graph of y = f⁻¹(x) is the reflection of y = f(x) in the line:
An inverse reflects the original graph in the line y = x; a point (a, b) becomes (b, a).
You've got it
Key idea
- domain = inputs, range = outputs; only one-one functions have an inverse
- composition: $fg(x) = f(g(x))$ (do $g$ first)
- $f^{-1}$: swap $x$ and $y$ and solve; its graph reflects $f$ in $y = x$