﻿ Domain Range Function Worksheet | Problems & Solutions

# Domain Range Function Worksheet

Domain Range Function Worksheet
• Page 1
1.
Find the domain of the function $\sqrt[4]{16-{x}^{2}}$.
 a. [- 4, 0) b. (- 4, 4] c. [- 4, 4] d. [0, 4] e. [4, ∞]

#### Solution:

f(x) = 16-x24
[Original function.]

Set 16 - x2 greater than or equal to zero to make the expression under the fourth root, positive.

16 - x2 ≥ 0, so x2 - 16 ≤ 0

(x - 4)(x + 4) ≤ 0
[Factor the left side of the inequality.]

x ≥ - 4 and x ≤ 4

So, the domain of the function is [- 4, 4].

2.
Find the domain of the function, $f$($x$) = - $\sqrt{-4x+5}$.
 a. All real values of $x$ b. All real values of $x$ ≤ - 1.25 c. All real values of $x$ ≥ 1.25 d. All real values of $x$ except 1.25 e. All real values of $x$ ≤ 1.25

#### Solution:

f(x) = - -4x+5
[Original function.]

f(x) is defined if - 4x + 5 ≥ 0

- 4x ≥ - 5
[Subtract 5 from both sides.]

4x ≤ 5
[Divide by - 1.]

x ≤ 1.25
[Divide by 4 on both sides.]

So, the domain is all real values of x less than or equal to 1.25.

3.
Find the domain of the function, $f$($x$) = $\frac{x+6}{{x}^{2}+10x+9}$.
 a. (- ∞, - 9) $\cup$ (- 1, ∞) b. [- 9, - 1] c. (- 9, - 1) $\cup$ (- 1, ∞) d. (- ∞, - 9) $\cup$ (- 9, - 1) e. (- ∞, - 9) $\cup$ (- 9, - 1) $\cup$ (- 1, ∞)

#### Solution:

f(x) = x+6x2+10x+9
[Original function.]

f (x) = x+6(x+9)(x+1)
[Factor the denominator.]

Set the denominator equal to zero to solve for x.

(x + 9) (x + 1) = 0

x = - 9, x = - 1

That is f(x) is not defined at x = - 9 and at x = - 1

So, the domain of the function is (- ∞, - 9) (- 9, - 1) (- 1, ∞).

4.
Find the domain: $f$($x$) = $\frac{2x+3}{|7x+6|}$
 a. (- $\frac{6}{7}$, 0) b. (- ∞, 0) $\cup$ (0, ∞) c. (- $\frac{6}{7}$, ∞) d. [- $\frac{6}{7}$, $\frac{6}{7}$] e. (- ∞, - $\frac{6}{7}$) U (- $\frac{6}{7}$, ∞)

#### Solution:

f(x) = 2x+3|7x+6|
[Original function.]

Set the denominator equal to zero to solve for x.

7x + 6 = 0

x = - 6 / 7

That is f(x) is not defined at x = - 6 / 7.

So, the domain of the function f is (- ∞, - 6 / 7) (- 6 / 7, ∞).

5.
Find the domain of the function $f$ ($x$) = $\frac{8{x}^{2}-3}{\sqrt{{x}^{2}-10}}$
 a. (- ∞, $\sqrt{10}$) $\cup$ ($\sqrt{10}$, ∞) b. (- ∞, ∞) c. (- ∞, 0) $\cup$ (0, ∞) d. [- $\sqrt{10}$, $\sqrt{10}$] e. (- ∞, - $\sqrt{10}$) $\cup$ ($\sqrt{10}$, ∞)

#### Solution:

f (x) = 8x2-3x2-10
[Original Function.]

Solve for x which makes the expression under the square root, positive.

x2 - 10 > 0

(x - 10)(x + 10) > 0
[Factor the left side of the inequality.]

x < - 10 or x > 10

So, the domain of the function is (- ∞, - 10) (10, ∞).

6.
Find the domain of the function $f$($x$) = $\sqrt{7x-8}$
 a. [$\frac{7}{8}$, ∞) b. [- $\frac{8}{7}$, ∞) c. [$\frac{8}{7}$, ∞) d. ($\frac{8}{7}$, ∞] e. ($\frac{8}{7}$, ∞)

#### Solution:

f(x) = 7x-8
[Original Function.]

Solve for x which makes the expression under the square root, positive.

7x - 8 ≥ 0

7x ≥ 8

x8 / 7
[Divide by 7 on both sides.]

So, the domain of the function is [8 / 7, ∞).

7.
Find the domain of the function: $y$ = $\frac{1}{\sqrt{2-x}}$
 a. [- ∞, 2] b. (2, ∞) c. [- ∞, 2) d. (- ∞, - 2) $\cup$ (- 2, ∞) e. (- ∞, 2)

#### Solution:

y = 12-x
[Original Function.]

Solve for x which makes the expression under the square root, positive.

2 - x > 0

2 > x

So, the domain of the function is (- ∞, 2).

8.
Find the domain: $f$ ($x$) = ${\left(11x-2\right)}^{\frac{5}{4}}$
 a. [$\frac{11}{2}$, ∞) b. [$\frac{2}{11}$, ∞] c. (- ∞, - $\frac{2}{11}$) d. [$\frac{2}{11}$, ∞) e. ($\frac{2}{11}$, ∞)

#### Solution:

f (x) = (11x-2)54
[Original Function.]

Solve for x which makes the expression under the fourth root, positive.

11x - 2 ≥ 0

x2 / 11

The domain of the function is [2 / 11, ∞).

9.
Find the domain of the function: $f$ ($x$) = $\frac{\sqrt{9-4x}}{\sqrt{3x-2}}$
 a. (- $\frac{2}{3}$, $\frac{9}{4}$) b. [$\frac{2}{3}$, $\frac{9}{4}$) c. ($\frac{2}{3}$, $\frac{9}{4}$] d. ($\frac{2}{3}$, $\frac{9}{4}$) e. [- $\frac{9}{4}$, - $\frac{2}{3}$]

#### Solution:

f (x) = 9-4x3x-2
[Original Function.]

Solve for x which makes the expression under the square root in the numerator, positive.

9 - 4x ≥ 0

x9 / 4

Solve for x which makes the expression under the square root in the denominator, positive.

3x - 2 > 0

x > 2 / 3

So, the domain of the function is (2 / 3, 9 / 4].

10.
What is the range of the function shown in the graph?

 a. [0, ∞) b. (- ∞, 4) $\cup$ (4, ∞) c. (0, ∞) d. (- ∞, 0) $\cup$ (0, ∞) e. [4, ∞]

#### Solution:

The y-coordinates of the points on the graph are all real numbers greater than or equal to zero.

So, the range of the function is [0, ∞).