One-to-One Mapping Math Example 4

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

Example 4

hard

Find the inverse of

h(x) = \dfrac{2x+1}{x-3}

and state its domain.

Solution

1
Set $y = \frac{2x+1}{x-3}$ , swap $x$ and $y$ : $x = \frac{2y+1}{y-3}$ .
2
Solve for $y$ : multiply both sides by $(y-3)$ : $x(y-3)=2y+1 \Rightarrow xy-3x=2y+1 \Rightarrow xy-2y=3x+1 \Rightarrow y(x-2)=3x+1 \Rightarrow y=\frac{3x+1}{x-2}$ .
3
So $h^{-1}(x)=\frac{3x+1}{x-2}$ . Domain: $x \neq 2$ (denominator restriction).

Answer

h^{-1}(x) = \dfrac{3x+1}{x-2}

, domain

x \neq 2

To find the inverse of a rational function, swap

x

and

y

then isolate

y

through algebraic manipulation. The domain of the inverse equals the range of the original function.

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.

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

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Related Concepts

Function Inverse Function Horizontal Line Test