Functions and Graphs: Master Functions and Their Visual Representations

Question 1

For the function f(x) = √(x - 2), what is its domain?

Accepted Answer

All real numbers greater than or equal to 2

Answer

All real numbers less than or equal to 2

Answer

All real numbers between 0 and 2

Answer

All real numbers

Question 2

To shift the graph of f(x) = x² left by 2 units, which transformation should be applied?

Accepted Answer

f(x - 2)

Answer

f(x + 2)

Answer

f(2x)

Answer

f(x)/2

Question 3

Consider the function f(x) = 1/(x - 2). Which of the following best describes its graph?

Accepted Answer

A hyperbola with a center at (2, 0)

Answer

A parabola with a vertex at (2, 0)

Answer

A circle with a center at (2, 0)

Answer

A line with a slope of 1 and a y-intercept of 2

Question 4

Determine the domain of the function f(x) = log(x - 1).

Accepted Answer

All real numbers greater than 1

Answer

All real numbers less than 1

Answer

All real numbers between 0 and 1

Answer

All real numbers

Question 5

Which of the following transformations will shift the graph of f(x) = x^2 up by 3 units?

Accepted Answer

f(x) + 3

Answer

f(x) - 3

Answer

f(x) + 3x

Answer

f(x) - 3|x|

Question 6

Find the amplitude of the function f(x) = 2cos(4x).

Accepted Answer

2

Answer

1

Answer

3

Answer

4

Question 7

Find the equation of the tangent line to the curve y = x^2 at the point (1, 1).

Accepted Answer

y = 2x

Answer

y = x

Answer

y = 3x

Answer

y = 4x

Question 8

The graph of the function f(x) = x^2 opens:

Accepted Answer

Upward

Answer

Downward

Answer

To the left

Answer

To the right

Question 9

The domain of the function f(x) = 1 / (x^2 - 4) is:

Accepted Answer

{x | x ≠ -2 and x ≠ 2}

Answer

All real numbers

Answer

{x | x < -2 or x > 2}

Answer

{x | -2 < x < 2}

Question 10

The graph of the function f(x) = cos(x) has a period of:

Accepted Answer

2π

Answer

π/2

Answer

π

Answer

4π

Question 11

Which property is NOT true for exponential functions?

Accepted Answer

Their graphs increase slowly

Answer

They are always positive

Answer

Their domain is all real numbers

Answer

Their range is all positive numbers

Question 12

The graph of the function f(x) = log(x) has a horizontal asymptote at:

Accepted Answer

y = 0

Answer

y = 1

Answer

x = 0

Answer

x = 1

Question 13

Which of the following functions is a polynomial function?

Accepted Answer

f(x) = x^2 + 2x - 3

Answer

f(x) = x^2 / (x - 1)

Answer

f(x) = e^x

Answer

f(x) = sin(x)

Question 14

Which of the following is the correct graph of the function f(x) = 2x - 1?

Accepted Answer

A straight line with a slope of 2 and a y-intercept of -1

Answer

A parabola with a vertex at (0, -1)

Answer

A circle with a center at (0, 1) and a radius of 2

Answer

An ellipse with foci at (0, 1) and (0, -1)

Question 15

Which of the following transformations would shift the graph of f(x) = x^2, 2 units to the right?

Accepted Answer

f(x + 2)

Answer

f(x - 2)

Answer

f(2x)

Answer

f(x/2)

Question 16

Find the domain of the function f(x) = 1/(x - 3).

Accepted Answer

All real numbers except 3

Answer

All real numbers except 0

Answer

All real numbers

Answer

All positive real numbers

Question 17

Find the range of the function f(x) = |x - 1|.

Accepted Answer

[0, ∞)

Answer

(0, ∞)

Answer

[-1, ∞)

Answer

(-1, ∞)

Question 18

The graph of the function f(x) has a minimum at (2, 5). Find the vertex of the graph.

Accepted Answer

(2, 5)

Answer

(-2, 5)

Answer

(5, 2)

Answer

(-5, 2)

Question 19

Find the equation of the line that is perpendicular to the graph of the function f(x) = 2x + 1 and passes through the point (3, 5).

Accepted Answer

y = -1/2x + 3.5

Answer

y = -x + 4

Answer

y = x + 4

Answer

y = 1/2x + 3.5

Question 20

Which of the following is the derivative of the function f(x) = e^x?

Accepted Answer

e^x

Answer

x e^x

Answer

1/e^x

Answer

ln(e^x)

Question 21

Which of the following graphs represents a linear function?

Accepted Answer

Straight line

Answer

Parabola

Answer

Ellipse

Answer

Circle

Question 22

Which transformation shifts the graph of f(x) left by 2 units?

Accepted Answer

f(x - 2)

Answer

f(x + 2)

Answer

f(2x)

Answer

f(x/2)

Question 23

Which of the following is an example of a polynomial function?

Accepted Answer

f(x) = x^3 + 2x^2 - 5x + 1

Answer

f(x) = 1/x

Answer

f(x) = e^x

Answer

f(x) = sin(x)

Question 24

What is the domain of the function f(x) = √(x - 3)?

Accepted Answer

(3, ∞)

Answer

[-3, ∞)

Answer

(∞, -3)

Answer

(3, 3)

Question 25

What is the period of the function f(x) = sin(2x)?

Accepted Answer

2π

Answer

π

Answer

π/2

Answer

1

Question 26

Which of the following is an example of a rational function?

Accepted Answer

f(x) = x/(x-1)

Answer

f(x) = x^2 + 1

Answer

f(x) = e^x

Answer

f(x) = sin(x)

Question 27

Which of the graphs represents a linear function?

Accepted Answer

Straight line

Answer

Parabola

Answer

Circle

Question 28

Determine the domain of the function f(x) = √x.

Accepted Answer

x ≥ 0

Answer

x < 0

Answer

x > 0

Answer

x ≤ 0

Question 29

Find the range of the function f(x) = -x^2 + 1.

Accepted Answer

y ≥ 1

Answer

y ≤ 1

Answer

y < 0

Answer

y > 0

Question 30

Write the equation of the line perpendicular to y = 2x + 1 and passing through the point (1, 3).

Accepted Answer

y = -1/2x + 2.5

Answer

y = 1/2x + 2.5

Answer

y = -2x + 4

Answer

y = 2x + 2

Question 31

A consulting company charges an initial consultation fee of $50, and then $20 for each additional hour of consulting. Write this relationship as a piecewise function.

Accepted Answer

f(x) = 50 if x = 0, 50 + 20x if x > 0

Answer

f(x) = 50 + 20x

Answer

f(x) = 20 if x = 0, 50 if x > 0

Answer

f(x) = 50x

Question 32

Which graph corresponds to the function f(x) = 2x + 1?

Accepted Answer

A straight line with a slope of 2 and a y-intercept of 1

Answer

A parabola with its vertex at (0, 1)

Answer

A circle with a radius of 1

Answer

An ellipse with a major axis of length 2

Question 33

What is the domain of the function g(x) = 1/x?

Accepted Answer

All real numbers except 0

Answer

All real numbers

Answer

All positive real numbers

Answer

All negative real numbers

Question 34

Which transformation shifts the graph of h(x) = x^2 to the left by 2 units?

Accepted Answer

h(x - 2)

Answer

h(x + 2)

Answer

h(2x)

Answer

h(x/2)

Question 35

Which of the following is a polynomial function?

Accepted Answer

f(x) = x^2 + 2x - 1

Answer

g(x) = e^x

Answer

h(x) = sin(x)

Answer

l(x) = log(x)

Question 36

Find the vertex of the parabola defined by the function f(x) = x^2 - 4x + 3.

Accepted Answer

(2, -1)

Answer

(1, 0)

Answer

(0, 1)

Answer

(2, 1)

Question 37

What is the range of the function h(x) = |x|?

Accepted Answer

All real numbers

Answer

All real numbers except 0

Answer

All positive real numbers

Answer

All negative real numbers

Question 38

Find the inverse of the function f(x) = 3x + 5.

Accepted Answer

f^(-1)(x) = (x - 5)/3

Answer

f^(-1)(x) = 3(x - 5)

Answer

f^(-1)(x) = (x - 3)/5

Answer

The function has no inverse.

Question 39

Identify the range of the function f(x) = |x|.

Accepted Answer

{y | y ≥ 0}

Answer

{y | y > 0}

Answer

{y | y < 0}

Answer

{y | y ∈ R}

Question 40

Which transformation shifts the graph of f(x) = x^2 three units to the left?

Accepted Answer

f(x - 3)

Answer

f(x + 3)

Answer

f(3x)

Answer

f(x/3)

Question 41

Which function exhibits periodicity?

Accepted Answer

f(x) = sin(x)

Answer

f(x) = x^2

Answer

f(x) = e^x

Answer

f(x) = |x|

Question 42

Identify the amplitude of the function f(x) = 2cos(3x).

Accepted Answer

2

Answer

1

Answer

3

Answer

6

Question 43

Which characteristic is NOT associated with functions?

Accepted Answer

Function graphs are exclusively linear.

Answer

Each input has only one output.

Answer

Domains and ranges of functions consist exclusively of real numbers.

Answer

Inverse functions are themselves functions.

Question 44

Identify the graph type that corresponds to the function f(x) = x^2 - 1.

Accepted Answer

Parabola opening upwards

Answer

Parabola opening downwards

Answer

Circular

Answer

Hyperbolic

Question 45

Which inverse function corresponds to f(x) = 2x + 1?

Accepted Answer

f^-1(x) = (x - 1) / 2

Answer

f^-1(x) = 2x - 1

Answer

f^-1(x) = x^2 - 1

Answer

f^-1(x) does not exist.

Question 46

Specify the range of the function f(x) = |x|.

Accepted Answer

y ≥ 0

Answer

y > 0

Answer

y < 0

Answer

y ≤ 0

Question 47

Determine the line equation perpendicular to f(x) = 2x + 1 passing through the point (1, 2).

Accepted Answer

y = -1/2x + 3/2

Answer

y = 1/2x + 3/2

Answer

y = -2x + 3

Answer

y = 2x + 1

Question 48

Determine the region's area bounded by f(x) = x^2 and the x-axis between x = 0 and x = 2.

Accepted Answer

4/3 square units

Answer

2/3 square units

Answer

8/3 square units

Answer

16/3 square units

Question 49

Which of the following is NOT a function transformation?

Accepted Answer

Addition

Answer

Translation

Answer

Reflection

Answer

Vertical stretch

Question 50

The graph of the linear function f(x) = 2x + 3 has a slope of:

Accepted Answer

2

Answer

1

Answer

3

Answer

4

Question 51

The inverse of the function f(x) = log₂(x) is:

Accepted Answer

f^-1(x) = 2^x

Answer

f^-1(x) = 1/log₂(x)

Answer

f^-1(x) = log₁₀(x)

Answer

f^-1(x) = e^x

Question 52

Which of the following functions is increasing on the interval (-∞, ∞)?

Accepted Answer

f(x) = e^x

Answer

f(x) = x^2

Answer

f(x) = sin(x)

Answer

f(x) = 1/x

Question 53

Which of the following is the graph of the quadratic function f(x) = x^2 + 2x - 3?

Accepted Answer

A parabola opening upwards

Answer

A parabola opening downwards

Answer

A straight line with a positive slope

Answer

A straight line with a negative slope

Question 54

What is the domain of the function f(x) = 1/(x-2)?

Accepted Answer

All real numbers except 2

Answer

All real numbers except 0

Answer

All real numbers

Answer

Only positive real numbers

Question 55

What is the inverse function of f(x) = 2x + 1?

Accepted Answer

f^(-1)(x) = (x-1)/2

Answer

f^(-1)(x) = x/2 - 1/2

Answer

f^(-1)(x) = 2x - 1

Answer

f^(-1)(x) does not exist

Question 56

Which of the following is a characteristic of a periodic function?

Accepted Answer

Its graph repeats itself at regular intervals

Answer

Its graph is always increasing or decreasing

Answer

Its domain is finite

Answer

Its range is infinite

Question 57

What is the amplitude of the function f(x) = 3sin(2x)?

Accepted Answer

3

Answer

1

Answer

2

Answer

-3