Chapter 1.2, Problem 7E

(a)

To determine

To find:The domain and range of the function $y=2+\sqrt{x-1}$ .

Answer to Problem 7E

The domain and range of the function $y=2+\sqrt{x-1}$ is $\left[1,\infty \right)$ and $\left[2,\infty \right)$ respectively.

Explanation of Solution

Given information:The given function is $y=2+\sqrt{x-1}$ .

Calculation:

The function $y=2+\sqrt{x-1}$ is defined only when $x-1\ge 0$ or $x\ge 1$ .So, the domain of the function $y=2+\sqrt{x-1}$ is the interval $\left[1,\infty \right)$ .

The function $\sqrt{x-1}$ is always non zero for the condition;

  $\sqrt{x-1}\ge 0$

Add $2$ to both the sides.

  $\begin{array}{c}2+\sqrt{x-1}\ge 0+2\\ y\ge 2\end{array}$

So, the range of the function $y=2+\sqrt{x-1}$ is the interval $\left[2,\infty \right)$ .

Therefore, the domain and range of the function $y=2+\sqrt{x-1}$ is $\left[1,\infty \right)$ and $\left[2,\infty \right)$ respectively.

(b)

To determine

To graph:The function $y=2+\sqrt{x-1}$ .

Explanation of Solution

Given information:The function is $y=2+\sqrt{x-1}$ .

Graph:

The graph of the function $y=2+\sqrt{x-1}$ is shown below:

  

Figure(1)

Interpretation:The graph of the function $y=2+\sqrt{x-1}$ is an rightwards parabola defined for only $x\ge 1$ .