Chapter 6.8, Problem 41PPSE

Solution

a.

To draw the graph.

Let us consider the parent function as:

  $f(x)=\sqrt{x}$

The first transformation is to shift the parent function in the horizontal direction towards right by $2$ units:

  $\begin{array}{l}g(x)=f(x-2)\\ \Rightarrow g(x)=\sqrt{x-2}\end{array}$

The second transformation is to vertically shift the graph by 2 units in downward direction:

  $\begin{array}{l}h(x)=g(x)-2\\ \Rightarrow h(x)=\sqrt{x-2}-2\end{array}$

Graph:The graph for the function is shown as:

  

b.

To determine the domain and the range.

  $\text{Domain}=\left[2,\infty \right)$$\text{Range}=\left[-2,\infty \right)$

Graph:The graph for the function is shown as:

  

Calculation:The graph is defined for all real numbers greater than $2$ . Hence, the domain is:

  $\text{Domain}=\left[2,\infty \right)$

Also, the graph covers all values of real numbers greater than $-2$ on $y-$ axis. Hence, the range is:

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

c.

To determine that function is onto the set of non-negative real numbers.

No

Graph:The graph for the function is shown as:

  

Calculation:The graph is defined for all real numbers greater than $2$ . Hence, the domain is:

  $\text{Domain}=\left\{2,\infty \right\}$

Also, the graph covers all values of real numbers greater than $-2$ on $y-$ axis. Hence, the range is:

  $\text{Range}=\left\{-2,\infty \right\}$

On the set of non-negative real numbers, function is not onto on the domain. For example, the number $2$ is in the domain, but there is no element mapped to it by the function in the set of non-negative real numbers.