Function as Mapping Examples in Math

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

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

Concept Recap

Viewing a function as a mapping means thinking of it as an explicit association from each element of the domain to exactly one element of the codomain.

Like a dictionary: every word maps to a definition. Every input maps to an 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: Functions are mappings between sets—input set to output set.

Common stuck point: A mapping must be well-defined: each input must produce exactly one output — a relation that maps one input to two outputs is not a function.

Sense of Study hint: Draw an arrow diagram: list inputs on the left, outputs on the right, and draw an arrow from each input to its output.

Worked Examples

Example 1

easy
Let f: \{1,2,3\} \to \{a,b,c\} be defined by f(1)=a, f(2)=a, f(3)=c. Determine whether f is a valid function, and find its range.

Solution

  1. 1
    A function requires every element of the domain to map to exactly one element of the codomain. Check: f(1)=a, f(2)=a, f(3)=c — each domain element has exactly one image. ✓ Valid function.
  2. 2
    The range is the set of actual output values: \{a, c\} (note b is in the codomain but not in the range).
  3. 3
    Observe this is many-to-one: both 1 and 2 map to a.

Answer

Valid function; range = \{a, c\}
A function maps each domain element to exactly one codomain element, but multiple domain elements may share an output (many-to-one). The range (image) is only the outputs actually achieved, which may be a proper subset of the codomain.

Example 2

medium
Explain why the relation R = \{(1,2),(1,3),(2,5)\} is NOT a function from \{1,2\} to \{2,3,5\}.

Practice Problems

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

Example 1

easy
Which of the following sets of ordered pairs defines a function from \{1,2,3\} to \mathbb{R}? (A) \{(1,5),(2,5),(3,5)\} (B) \{(1,2),(2,3)\} (C) \{(1,0),(2,1),(3,2),(1,4)\}

Example 2

medium
Let f: \mathbb{R} \to \mathbb{R}, f(x) = x^2. Find f^{-1}(\{4\}) (the pre-image of 4) and explain why f does not have an inverse function on all of \mathbb{R}.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

function definition