One-to-One Mapping Math Example 3

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

Example 3

easy

Which functions are one-to-one? (A)

f(x)=x^2

on

\mathbb{R}

. (B)

f(x)=e^x

. (C)

f(x)=|x|

.

Solution

1
(A) $f(x)=x^2$ : $f(-2)=f(2)=4$ , two inputs share one output. Not one-to-one.
2
(B) $f(x)=e^x$ : strictly increasing on all of $\mathbb{R}$ , so $a \neq b \Rightarrow e^a \neq e^b$ . One-to-one. ✓ (C) $f(x)=|x|$ : $f(-3)=f(3)=3$ . Not one-to-one.

Answer

Only (B)

f(x)=e^x

is one-to-one

Strictly monotone functions are always one-to-one. Even functions like

x^2

and

|x|

are symmetric about the

y

-axis, meaning opposite inputs give the same output, violating injectivity.

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

Determine whether [formula] is one-to-one by (a) the definition and (b) the horizontal line test.

Example 2 medium

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

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

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Related Concepts

Function Inverse Function Horizontal Line Test