Chapter 10.1, Problem 54E

(A)

To determine

To match the function and describe the range.

Answer to Problem 54E

Option B

Range = $[2,\infty )$

Explanation of Solution

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

Formula Used: Square root function cannot be negative.

Calculation:

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

So,

  $\sqrt{x}\ge 0$

Squaring both sides,

  $x\ge 0$

Hence,

Domain = $[0,\infty )$

As,

  $\sqrt{x}\ge 0$

Adding by 2 both sides

  $\sqrt{x}+2\ge 2$

Now, $f\left(x\right)=\sqrt{x}+2$

  $f\left(x\right)\ge 2$

Hence,

Range = $[2,\infty )$

The graph B has Domain = $[0,\infty )$ and Range = $[2,\infty )$ . So,

Option B is correct.

(B)

To determine

To match the function and describe the range.

Answer to Problem 54E

Option D

Range = $[-2,\infty )$

Explanation of Solution

Given information: $f\left(x\right)=\sqrt{x}+2$ and $m\left(x\right)=f\left(x\right)-4$

Formula Used: Square root function cannot be negative.

Calculation:

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

  $m\left(x\right)=f\left(x\right)-4$

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

  $m\left(x\right)=\sqrt{x}+2-4$

Now,

  $m\left(x\right)=\sqrt{x}-2$

So,

  $\sqrt{x}\ge 0$

Squaring both sides,

  $x\ge 0$

Hence,

Domain = $[0,\infty )$

As,

  $\sqrt{x}\ge 0$

Subtracting by 2 both sides

  $\sqrt{x}-2\ge -2$

Now, $m\left(x\right)=\sqrt{x}-2$

  $m\left(x\right)\ge -2$

Hence,

Range = $[-2,\infty )$

The graph D has Domain = $[0,\infty )$ and Range = $[-2,\infty )$ . So,

Option D is correct.

(C)

To determine

To match the function and describe the range.

Answer to Problem 54E

Option C

Range = $[2,\infty )$

Explanation of Solution

Given information: $f\left(x\right)=\sqrt{x}+2$ and $n\left(x\right)=f\left(-x\right)$

Formula Used: Square root function cannot be negative.

Calculation:

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

  $n\left(x\right)=f\left(-x\right)$

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

  $n\left(x\right)=\sqrt{-x}+2$

So,

  $\sqrt{x}\ge 0$

Squaring both sides,

  $x\ge 0$

Multiply by -1 both sides.

  $-x\ge 0$

So,

  $x\le 0$

Hence,

Domain = $(-\infty ,0]$

As,

  $\sqrt{x}\ge 0$

Squaring both sides

  $x\ge 0$

Multiply by -1 both sides.

  $-x\ge 0$

Square root both sides.

  $\sqrt{-x}\ge 0$

Adding by 2 both sides.

  $\sqrt{-x}+2\ge 2$

Now, $n\left(x\right)=\sqrt{-x}+2$

  $n\left(x\right)\ge 2$

Hence,

Range = $[2,\infty )$

The graph C has Domain = $[0,\infty )$ and Range = $[2,\infty )$ . So,

Option C is correct.

(D)

To determine

To match the function and describe the range.

Answer to Problem 54E

Option A

Range = $[2,\infty )$

Explanation of Solution

Given information: $f\left(x\right)=\sqrt{x}+2$ and $p\left(x\right)=f\left(3x\right)$

Formula Used: Square root function cannot be negative.

Calculation:

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

  $p\left(x\right)=f\left(3x\right)$

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

  $p\left(x\right)=\sqrt{3x}+2$

So,

  $\sqrt{x}\ge 0$

Squaring both sides,

  $x\ge 0$

Hence,

Domain = $[0,\infty )$

As,

  $\sqrt{x}\ge 0$

Squaring both sides

  $x\ge 0$

Multiply by 3 both sides.

  $3x\ge 0$

Square root both sides.

  $\sqrt{3x}\ge 0$

Adding by 2 both sides.

  $\sqrt{3x}+2\ge 2$

Now, $p\left(x\right)=\sqrt{3x}+2$

  $p\left(x\right)\ge 2$

Hence,

Range = $[2,\infty )$

The graph A has Domain = $[0,\infty )$ and Range = $[2,\infty )$ . So,

Option A is correct.