Chapter 1.5, Problem 14E

In Exercises 13−22, find a formula for $f^{-1}\left(x\right)$ . Give the domain of $f^{-1}$ , including any restrictions "inherited" from f.

14. $f\left(x\right)=2x+5$

Solution

To find: The inverse of $f\left(x\right)=2x+5$ and the domain of $f^{-1}$ .

  $f^{-1}\left(x\right)=\frac{x-5}{2}$ .

  $D\in \left(-\infty ,\infty \right)$ .

Given information:

  $f\left(x\right)=2x+5$

Concept Used:

Domain: It is input value of $x$ where function is well defined.

Calculation:

The domain of $f\left(x\right)=2x+5$ is all real numbers because it is a linear function.

Let $f\left(x\right)=y$

  $\begin{array}{c}y=2x+5\left[\text{switch }x\text{ and }y\right]\\ x=2y+5\left[\text{solve for }y\right]\\ x-5=2y\\ y=\frac{x-5}{2}\left[y=f^{-1}\left(x\right)\right]\end{array}$

The inverse of function is $f^{-1}\left(x\right)=\frac{x-5}{2}$ and the domain is all real number. $D\in \left(-\infty ,\infty \right)$ .