Functions
A function is a relation in which every member of the domain has exactly one image. It is written f(x), read "f of x". For f(x) = 2x + 3, f(4) = 11. A graph is a function if no vertical line cuts it more than once (the vertical line test).
Ways to show a function
| Representation | Example |
|---|---|
| Formula | f(x) = x² + 1 |
| Table | x: 0, 1, 2 and f(x): 1, 2, 5 |
| Arrow diagram | 0 → 1, 1 → 2, 2 → 5 |
| Graph | A parabola |
| Ordered pairs | (0, 1), (1, 2), (2, 5) |
Inverse of a function
The inverse f⁻¹ undoes f. To find it, write y = f(x), solve for x, then swap x and y. Its graph is the reflection of the graph of f in the line y = x. Only a one-to-one function has an inverse.
Worked exampleFinding an inverse
Find the inverse of f(x) = 2x + 3.
- Write y = 2x + 3.
- Make x the subject.
- Swap x and y.
Answer: f⁻¹(x) = (x − 3) ÷ 2. Check: f(f⁻¹(x)) = x.
Try itFunction and inverse
Choose f(x) = ax + b and an input x. The inverse brings you back.
Key terms
- Function
- A relation in which each x has exactly one image.
- Image
- The output f(x) for an input x.
- Inverse function
- A function that undoes another.
- Vertical line test
- A graph is a function if no vertical line cuts it twice.
Summary
- A function gives exactly one output for each input.
- It can be shown by formula, table, arrow diagram or graph.
- The inverse is found by solving for x and swapping.
- The graph of f⁻¹ is the reflection of f in y = x.