Chapter 3.5, Problem 31E

Solution

To find : The domain and range of the function without using graph $y=\frac{1}{2}\sqrt[3]{x+7}$ .

The domain is $x\ge -7$ and range is $y\ge 0$ .

Given information:

The given radical function is $y=\frac{1}{2}\sqrt[3]{x+7}$ .

Concept used:

The domain of a function  is a set of all possible values $x$ such that the function is defined.

The range of a function is a set of all possible values the function can take, when  $x$ is any number from the domain of the function.

If the radical function of the form

  $y=af\left(x-h\right)+k$ To find the domain set the function in equality $\ge 0$ and solve for $x$ and to find range look at the $k$ .

Calculations:

Here the function is $y=\frac{1}{2}\sqrt[3]{x+7}$ and it is a square root function and the value of the sure root should not be negative.

Set $x+7\ge 0\Rightarrow x\ge -7$

So the domain should $x\ge -7$ then when input values then all values of ranges are $y\ge 0$ .

Hence the domain is $x\ge -7$ and range is $y\ge 0$ .