Chapter 10, Problem 5CT

To determine

To graph the function and find Domain, Range.

Answer to Problem 5CT

Domain = $[3,\infty )$

Range = $[2,\infty )$

Explanation of Solution

Given information: $f\left(x\right)=\sqrt{x-3}+2;g\left(x\right)=\sqrt{x}$

Formula Used: Square root function cannot be negative.

Calculation:

  $f\left(x\right)=\sqrt{x-3}+2;g\left(x\right)=\sqrt{x}$

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

So,

  $\sqrt{x-3}\ge 0$

Now, squaring both sides,

  $x-3\ge 0$

So,

  $x\ge 3$

Hence,

Domain = $[3,\infty )$

As,

  $\sqrt{x-3}\ge 0$

Now, Add 2 both sides

  $\sqrt{x-3}+2\ge 2$

As, $f\left(x\right)=\sqrt{x-3}+2$

  $f\left(x\right)\ge 2$

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: $g\left(x\right)=\sqrt{x}$

Horizontal shift: right 3 units

Vertical shift: up 2 units

Stretch or shrink: none.