Chapter 2, Problem 59RE

To determine

To sketch: the graph of the function. Also state the domain, intercepts and range of the graph.

Answer to Problem 59RE

The intercepts are $\left(1,0\right)$ .

The domain is $\left\{x|x\ge 1\right\}$ or the interval $\left[1,\infty \right)$ .

The intercepts are $\left(1,0\right)$ , domain is $\left\{x|x\ge 1\right\}$ or the interval $\left[1,\infty \right)$ , and range is $\left\{y|y\ge 0\right\}$ or the interval $\left[0,\infty \right)$ .

Explanation of Solution

Given information:

Given function

  $h\left(x\right)=\sqrt{x-1}$

Calculation:

Graph the basic function $h\left(x\right)=\sqrt{x}$ .

  

Reflect the graph $h\left(x\right)=\sqrt{x-1}$ is obtained by shifting the graph of $h\left(x\right)=\sqrt{x}$ horizontally right 1 unit.

  

Intercepts are the points where the graph touches or crosses the coordinate axes. In $h\left(x\right)=\sqrt{x-1}$ , the graph touches the coordinate axes at $\left(1,0\right)$ .

So, the intercepts are $\left(1,0\right)$ .

The points on the graph have $x$ coordinates between 1 and $\infty$ , and for each number $x$ between 1 and $\infty$ , there is a point $\left(x,f\left(x\right)\right)$ on the graph. So, the domain is $\left\{x|x\ge 1\right\}$ or the interval $\left[1,\infty \right)$ .

The points on the graph have $y$ coordinates between $0$ and $\infty$ , and for each number $y$ , there is at least one number $x$ in the domain.

So, the range of $F\left(x\right)$ is $\left\{y|y\ge 0\right\}$ or the interval $\left[0,\infty \right)$ .

Therefore, the intercepts are $\left(1,0\right)$ , domain is $\left\{x|x\ge 1\right\}$ or the interval $\left[1,\infty \right)$ , and range is $\left\{y|y\ge 0\right\}$ or the interval $\left[0,\infty \right)$ .