Function Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Function.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

A function is a rule that assigns to each input in the domain exactly one output in the codomain — every input maps to precisely one output, never two.

A machine: put something in, get exactly one thing out. Same input always gives same output.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: One input \to one output. The vertical line test checks this on a graph.

Common stuck point: A circle is not a function (fails vertical line test)—one x gives two y values.

Sense of Study hint: Try the vertical line test: draw vertical lines across the graph and check if any line hits the curve more than once.

Worked Examples

Example 1

easy
Determine whether the relation \{(1, 3), (2, 5), (3, 7), (4, 9)\} is a function.

Solution

  1. 1
    A relation is a function if every input (first element) maps to exactly one output (second element).
  2. 2
    List the inputs: 1, 2, 3, 4. Each input appears exactly once, so each has a unique output.
  3. 3
    Since no input is repeated with a different output, this relation is a function.

Answer

\text{Yes, it is a function.}
A function assigns exactly one output to each input. The vertical line test for graphs and the uniqueness of first elements in ordered pairs are equivalent ways to check this.

Example 2

medium
Does the equation x^2 + y^2 = 25 define y as a function of x?

Example 3

medium
Given f(x) = 2x^2 - 3x + 1, find f(4) and determine if g = \{(1,2), (3,4), (1,5)\} is a function.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Is the relation \{(2, 4), (3, 6), (2, 8)\} a function? Explain.

Example 2

hard
A graph passes through the points (1, 2), (2, 5), (3, 5), and (1, 7). Does this graph represent a function? Explain using the vertical line test.
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