Chapter SH, Problem 6.9EP

To determine

Find the domain and range of the given function.

Answer to Problem 6.9EP

The domain and the range of the function is given below.

Domain $=\left[1,\infty \right)$

Range $=\left[-2,\infty \right)$

Explanation of Solution

Given:

The given functionis $f\left(x\right)=\sqrt{x-1}-2$ .

Calculation:

The domain is the set of input values for which the function is real and defined.

  $f\left(x\right)=\sqrt{x-1}-2$

The domain only includes values for which the radicand is nonnegative.

  $x-1\ge 0$

Add $1$ to both sides.

  $\begin{array}{l}\Rightarrow x-1+1\ge 0+1\\ \Rightarrow x\ge 1\end{array}$

Domain $=\left[1,\infty \right)$

The range of the radical function $f\left(x\right)=p\sqrt{qx+r}+t$ is $f\left(x\right)\ge t$ .

Range of the function is $f\left(x\right)\ge -2$ .

Range $=\left[-2,\infty \right)$

Hence the domain and the range of the function is given below.

Domain $=\left[1,\infty \right)$

Range $=\left[-2,\infty \right)$