Chapter 10, Problem 4CR

To determine

To graph the function and find Domain, Range.

Answer to Problem 4CR

Domain = $[6,\infty )$

Range = $[2,\infty )$

Explanation of Solution

Given information: $g\left(x\right)=\frac{1}{4}\sqrt{x-6}+2$

Formula Used: Square root function cannot be negative.

Calculation:

  $g\left(x\right)=\frac{1}{4}\sqrt{x-6}+2$

As the domain of square root function is greater than or equal to zero i.e. cannot be negative.

So,

  $\sqrt{x-6}\ge 0$

Squaring both sides,

  $x-6\ge 0$

Now,

  $x\ge 6$

Hence,

Domain = $[6,\infty )$

As,

  $\sqrt{x-6}\ge 0$

Now, Division by 4 both sides

  $\frac{1}{4}\sqrt{x-6}\ge 0$

So, Add 2 both sides

  $\frac{1}{4}\sqrt{x-6}+2\ge 0+2$

So,

  $\frac{1}{4}\sqrt{x-6}+2\ge 2$

As, $g\left(x\right)=\frac{1}{4}\sqrt{x-6}+2$

  $g\left(x\right)\ge 2$

Range = $[2,\infty )$

To find the transformation, compare the function to the parent function and check to see if there is a horizontal or vertical shift, reflection about the x-axis or y-axis.

Parent function: $f\left(x\right)=\sqrt{x}$

Horizontal shift: right 6 units

Vertical shift: up 2 units

Stretch or shrink: shrink.