Chapter 3.5, Problem 33E

Solution

To find : The domain and range of the function without using graph $y=\frac{1}{4}\sqrt{x-3}+6$ .

The domain is $x\ge 3$ and range is $y\ge 6$ .

Given information:

The given radical function is $y=\frac{1}{4}\sqrt{x-3}+6$ .

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 $y=\frac{1}{4}\sqrt{x-3}+6$ and it is a square root function and the value of the sure root should not be negative.

Set $x-3\ge 0\Rightarrow x\ge 3$

So the domain should $x\ge 3$ then when input values then all values of ranges are $y\ge 6$ .

Hence the domain is $x\ge 3$ and range is $y\ge 6$ .