Exercise 1

Determine the domain of the following polynomial functions:

1 f(x) = 2 { x }^{ 5 } - 6 { x }^{ 3 } + 8 { x }^{ 2 } - 5

2 f(x) = \frac { 2 { x }^{ 2 } - 3 }{ 5 }

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Exercise 2

Determine the domain of the following rational functions:

1 f(x) = \frac { 2 { x }^{ 2 } - 3 }{ x + 2 }

2 f(x) = \frac { 2 { x }^{ 2 } - 3 }{ { x }^{ 2 } - 1 }

3 f(x) = \frac { 2 { x }^{ 2 } - 3 }{ { x }^{ 2 } + 1 }

4 f(x) = \frac { 2 { x }^{ 2 } - 3 }{ { x }^{ 2 } +2x + 1 }

5 f(x) = \frac { 2 { x }^{ 2 } - 3 }{ { x }^{ 3 } + 3 { x }^{ 2 } + 3x + 1 }

Exercise 3

Determine the domain of the following radical functions:

1 f(x) = \sqrt { x - 2 }

2 f(x) = \sqrt { - x + 2 }

3 f(x) = \sqrt { { x }^{ 2 } - 6x + 8 }

4 f(x) = \sqrt { - { x }^{ 2 } + 6x - 8 }

5 f(x) = \sqrt { { x }^{ 2 } + 4x + 4 }

6 f(x) = \sqrt { { x }^{ 2 } + x + 4 }

7 f(x) = \sqrt { - { x }^{ 2 } - 4x - 4 }

8 f(x) = \sqrt { { x }^{ 3 } - 4 { x }^{ 2 } + 3x }

9 f(x) = \frac { x - 5 }{ \sqrt { x - 2 } }

10 f(x) = \frac { \sqrt { x - 2 } }{ x - 5 }

11 f(x) = \sqrt[ 3 ]{ \frac { 3x + 2 }{ x + 1 } }

Exercise 4

Determine the domain of the following exponential functions:

1  f(x) = { e }^{ 2x - 3 }

2 f(x) = { e }^{ \frac { 2x - 3 }{ x } }

Exercise 5

Determine the domain of the following logarithmic functions:

1  f(x) = ln (x - 2)

2 f(x) = ln ( \frac { x }{ { x }^{ 2 } + 1 } )

Exercise 6

Determine the domain of the following trigonometric functions:

1 f(x) = \sqrt { 1 - \sin^{ 2 }{ x } }

2 f(x) = \sqrt { 1 - \cos { x } }

Exercise 7

Determine the domain of the following functions:

1  f(x) = \frac { 2 { x }^{ 2 } - 3 }{ ( { x }^{ 2 } - 9 )( { x }^{ 2 } - 4 )}

2  f(x) = \left\{\begin{matrix} \frac { x }{ x + 4 } \qquad if \quad x < 0 \\ \sqrt { \frac { 3 }{ x - 3 } } \qquad if \quad x >0 \end{matrix}\right

 

 

Solution of exercise 1

Determine the domain of the following polynomial functions:

1 f(x) = 2 { x }^{ 5 } - 6 { x }^{ 3 } + 8 { x }^{ 2 } - 5

D = \mathbb{R}

 

2 f(x) = \frac { 2 { x }^{ 2 } - 3 }{ 5 }

D = \mathbb{R}

 

Solution of exercise 2

Determine the domain of the following rational functions:

1 f(x) = \frac { 2 { x }^{ 2 } - 3 }{ x + 2 }

x + 2 = 0; \qquad D = \mathbb{R} - \left \{ -2 \right \}

 

2 f(x) = \frac { 2 { x }^{ 2 } - 3 }{ { x }^{ 2 } - 1 }

{ x }^{ 2 } - 1 = 0; \qquad D = \mathbb{R} - \left \{ -1, 1 \right \}

 

3 f(x) = \frac { 2 { x }^{ 2 } - 3 }{ { x }^{ 2 } + 1 }

{ x }^{ 2 } + 1 = 0; \qquad D = \mathbb{R}

 

4 f(x) = \frac { 2 { x }^{ 2 } - 3 }{ { x }^{ 2 } +2x + 1 }

{ x }^{ 2 } +2x + 1 = 0; \qquad { (x + 1) }^{ 2 } = 0 \qquad D = \mathbb{R} - \left \{ -1 \right \}

 

5 f(x) = \frac { 2 { x }^{ 2 } - 3 }{ { x }^{ 3 } + 3 { x }^{ 2 } + 3x + 1 }

{ x }^{ 3 } + 3 { x }^{ 2 } + 3x + 1 = 0; \qquad { (x + 1) }^{ 3 } = 0 \qquad D = \mathbb{R} - \left \{ -1 \right \}

 

Solution of exercise 3

Determine the domain of the following radical functions:

1 f(x) = \sqrt { x - 2 }

x - 2 \geq 0 \qquad D = [2, \infty)

 

2 f(x) = \sqrt { - x + 2 }

- x + 2 \geq 0 \qquad D = ( - \infty, 2 ]

 

3 f(x) = \sqrt { { x }^{ 2 } - 6x + 8 }

{ x }^{ 2 } - 6x + 8 \geq 0 \qquad D = ( - \infty, 2 ] U [ 4, \infty )

 

4 f(x) = \sqrt { - { x }^{ 2 } + 6x - 8 }

- { x }^{ 2 } + 6x - 8 \geq 0 \qquad D = [2, 4]

 

5 f(x) = \sqrt { { x }^{ 2 } + 4x + 4 }

{ (x + 2) }^{ 2 } \geq 0 \qquad D = \mathbb{R}

 

6 f(x) = \sqrt { { x }^{ 2 } + x + 4 }

{ x }^{ 2 } + x + 4 \geq 0 \qquad D = \mathbb{R}

 

7 f(x) = \sqrt { - { x }^{ 2 } - 4x - 4 }

- { (x + 2) }^{ 2 } \geq 0 \qquad D = -2

 

8 f(x) = \sqrt { { x }^{ 3 } - 4 { x }^{ 2 } + 3x }

x ({ x }^{ 2 } - 4x + 3) \geq 0 \qquad x (x - 1)(x - 3) \geq 0

D = [0, 1] U [3, \infty)

 

9 f(x) = \frac { x - 5 }{ \sqrt { x - 2 } }

x - 2 > 0 \qquad D = (2, \infty)

 

10 f(x) = \frac { \sqrt { x - 2 } }{ x - 5 }

\left\{\begin{matrix} x - 5 \neq 0 \qquad D = \mathbb{R} - \left \{ 5 \right \}\\ x - 2 \geq 0 \qquad D = [2, \infty) \end{matrix}\right \qquad D =[2, 5) U (5, \infty)

 

11 f(x) = \sqrt[ 3 ]{ \frac { 3x + 2 }{ x + 1 } }

D = \mathbb{R} - \left \{ -1 \right \}

 

Solution of exercise 4

 Determine the domain of the following exponential functions:

1  f(x) = { e }^{ 2x - 3 }

D = \mathbb{R}

2 f(x) = { e }^{ \frac { 2x - 3 }{ x } }

D = \mathbb{R} - \left \{ 0 \right \}

 

Solution of exercise 5

 Determine the domain of the following logarithmic functions:

1  f(x) = ln (x - 2)

x - 2 > 0 \qquad D = (2, \infty)

2 f(x) = ln ( \frac { x }{ { x }^{ 2 } + 1 } )

\frac { x }{ { x }^{ 2 } + 1 } > 0 \qquad x > 0 \qquad D = (0, \infty)

 

Solution of exercise 6

Determine the domain of the following trigonometric functions:

1 f(x) = \sqrt { 1 - \sin^{ 2 }{ x } }

1 - \sin^{ 2 }{ x } \geq 0 \qquad \sin^{ 2 }{ x } \leq 1 \qquad D = \mathbb{R}

2 f(x) = \sqrt { 1 - \cos { x } }

1 - \cos { x } \geq 0 \qquad \cos { x } \leq 1 \qquad D = \mathbb{R}

 

Solution of exercise 7

Determine the domain of the following functions:

1  f(x) = \frac { 2 { x }^{ 2 } - 3 }{ ( { x }^{ 2 } - 9 )( { x }^{ 2 } - 4 )}

( { x }^{ 2 } - 9 )( { x }^{ 2 } - 4 ) = 0; \qquad D = \mathbb{R} - \left \{ -3, -2, 2, 3 \right \}

 

2  f(x) = \left\{\begin{matrix} \frac { x }{ x + 4 } \qquad if \quad x < 0 \\ \sqrt { \frac { 3 }{ x - 3 } } \qquad if \quad x >0 \end{matrix}\right

\left\{\begin{matrix} x + 4 \neq 0 \\ x - 3 > 0 \end{matrix}\right \qquad D = ( - \infty, -4 ) U ( -4, 3 ) U ( 3, \infty)

 

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Hamza

Hi! I am Hamza and I am from Pakistan. My hobbies are reading, writing and playing chess. Currently, I am a student enrolled in the Chemical Engineering Bachelor program.