One-to-One Mapping Math Example 2

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

Example 2

medium

Show that

g(x) = x^3

is one-to-one on

\mathbb{R}

, then find its inverse function.

Solution

1
One-to-one proof: suppose $g(a)=g(b)$ , i.e., $a^3=b^3$ . Taking cube roots (valid over all reals): $a=b$ . So $g$ is one-to-one. ✓
2
Find inverse: swap $x$ and $y$ in $y=x^3$ , giving $x=y^3$ . Solve for $y$ : $y = x^{1/3} = \sqrt[3]{x}$ .
3
So $g^{-1}(x) = \sqrt[3]{x}$ . Verify: $g(g^{-1}(x))= (\sqrt[3]{x})^3 = x$ ✓ and $g^{-1}(g(x))=\sqrt[3]{x^3}=x$ ✓.

Answer

g^{-1}(x) = \sqrt[3]{x}

One-to-one functions have inverses. To find the inverse algebraically, swap variables and solve. Cube root is the inverse of cubing because both are odd functions defined on all of

\mathbb{R}

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 1 easy

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

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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Related Concepts

Function Inverse Function Horizontal Line Test