One-to-One Mapping Math Example 1

Follow the full solution, then compare it with the other examples linked below.

Example 1

easy
Determine whether f(x)=2x+3f(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)f(a)=f(b). Then 2a+3=2b+3⇒2a=2b⇒a=b2a+3=2b+3 \Rightarrow 2a=2b \Rightarrow a=b. So f(a)=f(b)⇒a=bf(a)=f(b) \Rightarrow a=b. ff is one-to-one. ✓
  2. 2
    Horizontal line test: f(x)=2x+3f(x)=2x+3 is a line with positive slope. Any horizontal line y=cy=c intersects it at exactly one point (x=(c−3)/2)(x = (c-3)/2).
  3. 3
    Both methods confirm ff is one-to-one (injective).

Answer

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

About One-to-One Mapping

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

Learn more about One-to-One Mapping →

More One-to-One Mapping Examples

Example 2 medium

Show that [formula] is one-to-one on [formula], then find its inverse function.

Example 3 easy

Which functions are one-to-one? (A) [formula] on [formula]. (B) [formula]. (C) [formula].

Example 4 hard

Find the inverse of [formula] and state its domain.

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