Chapter 3.5, Problem 21E

Solution

To find : Domain and range from the graph of $h\left(x\right)=-3\sqrt[3]{x+7}-6$ .

Domain $=\left\{x\in R,x\ge -7\right\}$ and range $=\left\{y\in R,y\le -6\right\}$ .

Given information:

The given function is $h\left(x\right)=-3\sqrt[3]{x+7}-6$ .

Concept used:

If graph $y=af\left(x-h\right)+k$ is the transformed graph of parent function $y=f\left(x\right)$ then $k$ is positive then graph is vertical upward and $k$ is negative the graph will be vertically downwards by the factor $k$ .Also if h is positive graph translated to right and negative it will translated to left by the factor h. Similarly if $a$ is positive it is vertically stretched by the factor $a$ and negative it will vertical stretched and reflected to $x$ axis.

Explanations:

Sketch graph and then find the range and domain.

The graph of the function is given below.

  

Here the function is $h\left(x\right)=-3\sqrt[3]{x+7}-6$ and $k=-6$ .then graph is vertically down wards by the factor $k=6$ and horizontally translated left by the factor $h=-7$ and reflected to $x$ axis.

Here will become domain $\left\{x\in R,x\ge -7\right\}$ and range will be $\left\{y\in R,y\le -6\right\}$ .