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Functions

12 min read

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).

A quadratic graphThe graph of y equals x squared minus 2x minus 3. It is a U-shaped curve called a parabola. It crosses the x axis at x equals minus 1 and x equals 3, has its lowest point, the vertex, at 1 and minus 4, and is symmetrical about the line x equals 1.xyaxis of symmetry x = 1(−1, 0)(3, 0)vertex (1, −4)(0, −3)
A straight line: each x has just one y.

Ways to show a function

RepresentationExample
Formulaf(x) = x² + 1
Tablex: 0, 1, 2 and f(x): 1, 2, 5
Arrow diagram0 → 1, 1 → 2, 2 → 5
GraphA 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.

  1. Write y = 2x + 3.
  2. Make x the subject.
  3. 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.

Check yourself If f(x) = x² − 1, f(3) =
Check yourself The inverse of f(x) = x + 7 is...
Check yourself A graph is a function if...

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.
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