Calculate the X- and Y-Intercepts of the Following Functions:

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Exercise 1

f(x) = 3x - { x }^{ 3 }

Exercise 2

f(x) = { x }^{ 4 } - 2 { x }^{ 2 } - 8

Exercise 3

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

Exercise 4

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

Exercise 5

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

Exercise 6

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

Exercise 7

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

Exercise 8

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

Exercise 9

f(x) = (x - 1) { e }^{ -x }

Exercise 10

f(x) = \frac { 1 }{ 2 \sqrt { 2 \pi } } { e }^{ - \frac { 1 }{ 2 } { x }^{ 2 } }

Exercise 11

f(x) = \frac { \ln { x } }{ x }

 

 

Solution of exercise 1

f(x) = 3x - { x }^{ 3 }

x-intercepts:

3x - { x }^{ 3 } = 0 \qquad x = \pm \sqrt { 3 } \qquad x = 0

(- \sqrt { 3 }, 0) \qquad (0,0) \qquad (\sqrt { 3 }, 0)

y-intercept:

f(0) = 3(0) - { (0) }^{ 3 } = 0

 

 

Solution of exercise 2

f(x) = { x }^{ 4 } - 2 { x }^{ 2 } - 8

x-intercepts:

{ x }^{ 4 } - 2 { x }^{ 2 } - 8 = 0 \qquad x = \pm 2

(-2, 0) \qquad  (2, 0)

y-intercept:

(0, -8)

 

Solution of exercise 3

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

x-intercepts:

\frac { { x }^{ 3 } }{ { (x - 1) }^{ 2 } } = 0 \qquad (0,0)

y-intercept:

(0,0)

 

Solution of exercise 4

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

x-intercepts:

\frac { { x }^{ 4 } + 1 }{ { x }^{ 2 } } = 0 \qquad { x }^{ 4 } + 1 = 0 \qquad x = \pm \sqrt[4]{ -1 }

No x-intercepts.

y-intercept:

f(0) = \frac { { (0) }^{ 4 } + 1 }{ { (0) }^{ 2 } } = \frac { 1 }{ 0 }

No y-intercept.

 

Solution of exercise 5

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

x-intercepts:

\frac { { x }^{ 2 } }{ 2 - x } = 0 \qquad (0,0)

y-intercept:

(0,0)

 

Solution of exercise 6

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

x-intercept:

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

y-intercept:

f(0) = \frac { 0 }{ 1 + { 0 }^{ 2 } } = \frac { 1 }{ 0 } = 0

(0,0)

 

Solution of exercise 7

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

x-intercepts:

\frac { { x }^{ 2 } - 3x + 2 }{ { x }^{ 2 } + 1 } = 0 \qquad { x }^{ 2 } - 3x + 2 = 0 \qquad x = 2 \quad x = 1

(2, 0) \qquad (1, 0)

y-intercept:

f(0) = \frac { { 0 }^{ 2 } - 3(0) + 2 }{ { 0 }^{ 2 } + 1 } = 2 \qquad (0, 2)

 

Solution of exercise 8

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

x-intercepts:

x + \sqrt { x } = 0 \qquad x = 0

(0,0)

y-intercept:

(0,0)

 

Solution of exercise 9

f(x) = (x - 1) { e }^{ -x }

x-intercepts:

(x - 1) { e }^{ -x } = 0 \qquad (1, 0)

y-intercept:

f(0) = (0 - 1) { e }^{ -(0) } \qquad (0, -1)

 

Solution of exercise 10

f(x) = \frac { 1 }{ 2 \sqrt { 2 \pi } } { e }^{ - \frac { 1 }{ 2 } { x }^{ 2 } }

x-intercepts:

\frac { 1 }{ 2 \sqrt { 2 \pi } } { e }^{ - \frac { 1 }{ 2 } { x }^{ 2 } } = 0 \qquad { e }^{ - \frac { 1 }{ 2 } { x }^{ 2 } } = 0

When we will take \ln on both sides, the right-hand-side will be equal to infinity hence no x-intercept.

y-intercept:

f(0) = \frac { 1 }{ 2 \sqrt { 2 \pi } } { e }^{ - \frac { 1 }{ 2 } { 0 }^{ 2 } } = \frac { 1 }{ 2 \sqrt { 2 \pi } } \qquad (0, \frac { 1 }{ 2 \sqrt { 2 \pi } })

 

Solution of exercise 11

f(x) = \frac { \ln { x } }{ x }

x-intercepts:

\frac { \ln { x } }{ x } = 0 \qquad \ln { x } = 0 \qquad { e }^{ 0 } = x

(1, 0)

y-intercept:

f(0) = \frac { \ln { 0 } }{ 0 }

No y-intercept

 

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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.