One-to-One Mapping Examples in Math

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

A one-to-one (injective) function maps every distinct input to a distinct output — no two different inputs produce the same output.

No two inputs share the same output—like social security numbers.

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-to-one (injective) functions have unique outputs for each input.

Common stuck point: Test: horizontal line hits graph at most once \to one-to-one.

Sense of Study hint: Try the horizontal line test: slide a horizontal line up and down the graph. If it ever crosses more than once, the function is not one-to-one.

Worked Examples

Example 1

easy
Determine whether f(x) = 2x + 3 is one-to-one by (a) the definition and (b) the horizontal line test.

Solution

  1. 1
    Definition: assume f(a)=f(b). Then 2a+3=2b+3 \Rightarrow 2a=2b \Rightarrow a=b. So f(a)=f(b) \Rightarrow a=b. f is one-to-one. ✓
  2. 2
    Horizontal line test: f(x)=2x+3 is a line with positive slope. Any horizontal line y=c intersects it at exactly one point (x = (c-3)/2).
  3. 3
    Both methods confirm f is one-to-one (injective).

Answer

f(x)=2x+3 is one-to-one
A one-to-one function has no two distinct inputs sharing an output: f(a)=f(b)\Rightarrow a=b. Linear functions with non-zero slope are always one-to-one because they are strictly monotone.

Example 2

medium
Show that g(x) = x^3 is one-to-one on \mathbb{R}, then find its inverse function.

Practice Problems

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

Example 1

easy
Which functions are one-to-one? (A) f(x)=x^2 on \mathbb{R}. (B) f(x)=e^x. (C) f(x)=|x|.

Example 2

hard
Find the inverse of h(x) = \dfrac{2x+1}{x-3} and state its domain.
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Background Knowledge

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

function definition