Functions: Master the Fundamentals of Mathematical Relationships

Question 1

Identify the transformation that will shift the graph of y = x^2 two units to the left.

Accepted Answer

y = (x-2)^2

Answer

y = (x+2)^2

Answer

y = x^2 - 2

Answer

y = x^2 + 2

Question 2

Find the inverse of the function f(x) = 2x + 3.

Accepted Answer

f^-1(x) = (x - 3)/2

Answer

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

Answer

f^-1(x) = 1/2x - 3/2

Answer

The inverse of f(x) does not exist.

Question 3

Identify an example of a composite function.

Accepted Answer

h(x) = f(g(x))

Answer

f(x) = x^2 + 1

Answer

g(x) = sin(x) + cos(x)

Answer

j(x) = e^x

Question 4

Identify the function that is increasing over the interval (-∞, ∞).

Accepted Answer

h(x) = e^x

Answer

f(x) = x^2

Answer

g(x) = sin(x)

Answer

j(x) = 1/x

Question 5

What is the inverse of y = 2x + 1?

Accepted Answer

y = (x-1)/2

Answer

y = 1/2x - 1

Answer

y = 2x - 1

Answer

y = -2x + 1

Question 6

Which is an example of a piecewise function?

Accepted Answer

f(x) = x^2 if x ≥ 0, -x^2 if x < 0

Answer

f(x) = 2x + 1

Answer

f(x) = sin(x)

Answer

f(x) = e^x

Question 7

Find the domain of the function f(x) = sqrt(x - 2) given in interval notation

Accepted Answer

(2, ∞)

Answer

[2, ∞)

Answer

[-∞, 2]

Answer

(-∞, 2]

Question 8

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

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 9

Find the composite function (f ∘ g)(x) where f(x) = x^3 and g(x) = x + 2

Accepted Answer

x^3 + 6x + 8

Answer

x^3 + 2

Answer

x^3 + 8

Answer

(x + 2)^3

Question 10

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

Accepted Answer

It is an increasing function

Answer

It is a decreasing function

Answer

It has a minimum at x = 0

Answer

It has a maximum at x = 0

Question 11

Identify a property of exponential functions.

Accepted Answer

They have a horizontal asymptote.

Answer

They always increase.

Answer

They pass through the origin.

Answer

They have a vertical asymptote.

Question 12

Write the equation of the exponential function with a y-intercept of 3 and an asymptote at y = 0.

Accepted Answer

y = 3e^x

Answer

y = 3 - e^x

Answer

y = 3 + e^x

Answer

y = 3/e^x

Question 13

Which of the following is a trigonometric function?

Accepted Answer

Sine

Answer

Logarithm

Answer

Exponential

Answer

Polynomial

Question 14

Find the amplitude of the trigonometric function y = 2cos(3x).

Accepted Answer

2

Answer

1

Answer

3

Answer

6

Question 15

Which of the following is NOT a characteristic of a well-defined function?

Accepted Answer

The graph of the function is a straight line.

Answer

Each input has exactly one output.

Answer

The output depends on the input.

Answer

The input can be any real number.

Question 16

Find the domain of the function f(x) = 1 / (x - 2) which represents the set of all possible values for the independent variable x.

Accepted Answer

All real numbers except 2

Answer

All real numbers

Answer

All real numbers greater than 2

Answer

All real numbers less than 2

Question 17

Evaluate f(g(x)) if f(x) = x^2 and g(x) = x + 1.

Accepted Answer

x^2 + 2x + 1

Answer

x^2 + 1

Answer

x^2 - 1

Answer

(x + 1)^2

Question 18

Find the period of the function f(x) = sin(2x), which represents the distance between repeating patterns on the graph of the function.

Accepted Answer

2π

Answer

π

Answer

π/2

Answer

2π/3

Question 19

Find the equation of the tangent line to the graph of f(x) = e^x at the point (0, 1). A tangent line touches a curve at a single point and has the same slope as the curve at that point.

Accepted Answer

y = x + 1

Answer

y = x

Answer

y = 2x + 1

Answer

y = 2x

Question 20

Determine the domain of the function f(x) = √(x - 2).

Accepted Answer

[2, ∞)

Answer

(−∞, ∞)

Answer

[0, ∞)

Answer

(-2, ∞)

Question 21

Given f(x) = sin(x) and g(x) = cos(x), determine (f∘g)(x).

Accepted Answer

sin(cos(x))

Answer

cos(sin(x))

Answer

-sin(cos(x))

Answer

-cos(sin(x))

Question 22

Which transformation will shift the graph of f(x) = x² to the right by 3 units?

Accepted Answer

f(x + 3)

Answer

f(x - 3)

Answer

f(3x)

Answer

f(x/3)

Question 23

Which of the following accurately represents the domain of a function f(x)?

Accepted Answer

{x | f(x) is defined}

Answer

{y | f(y) is defined}

Answer

{x | f(x) is undefined}

Answer

{y | f(y) is undefined}

Question 24

To shift the graph of the function f(x) = x^2 - 4 upwards by 2 units, which transformation should be applied?

Accepted Answer

f(x) + 2

Answer

f(x + 2)

Answer

f(x) - 2

Answer

f(x - 2)

Question 25

Which of the following functions is a composite function?

Accepted Answer

f(x) = g(h(x))

Answer

f(x) = cos(x)

Answer

f(x) = x^3

Answer

f(x) = ln(x)

Question 26

Which of the following is a characteristic of rational functions?

Accepted Answer

Can have both vertical and horizontal asymptotes

Answer

Always continuous over their domain

Answer

Always increasing

Answer

Always decreasing

Question 27

Find the zeros of the function f(x) = x^2 - 5x + 6.

Accepted Answer

x = 2 and x = 3

Answer

x = -2 and x = -3

Answer

x = 3 and x = -2

Answer

x = -3 and x = 2

Question 28

Which of the following equations represents a straight line with a slope of -2 and a y-intercept of 3?

Accepted Answer

f(x) = -2x + 3

Answer

f(x) = -2x - 3

Answer

f(x) = 2x + 3

Answer

f(x) = 2x - 3

Question 29

Which of the following is a composite function?

Accepted Answer

f(x) = sin(x²)

Answer

f(x) = x² + 3

Answer

f(x) = |x| - 2

Answer

f(x) = √x + 1

Question 30

Which graph represents the function y = |x| + 2?

Accepted Answer

A V-shaped graph with its vertex at (0, 2)

Answer

A circle centered at (0, 2) with radius 2

Answer

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

Answer

A parabola opening upwards with its vertex at (0, 2)

Question 31

Find the zeros of the quadratic function f(x) = x² - 9.

Accepted Answer

x = ±3

Answer

x = 9

Answer

x = 0

Answer

x = -9

Question 32

Which of the following is NOT a characteristic of a function?

Accepted Answer

Every input has multiple outputs

Answer

The input and output are related by a rule

Answer

Every output has one input

Answer

The domain and range are sets

Question 33

Find the domain of the function f(x) = sqrt(x - 2):

Accepted Answer

x ≥ 2

Answer

All real numbers

Answer

2 < x < ∞

Answer

-∞ < x < 2

Question 34

Which transformation maps the graph of f(x) to g(x) = f(x + 3)?

Accepted Answer

Translation 3 units to the left

Answer

Vertical stretch by a factor of 3

Answer

Translation 3 units up

Answer

Reflection about the x-axis

Question 35

Find the inverse function of f(x) = 2x + 1:

Accepted Answer

g(x) = (x - 1) / 2

Answer

g(x) = x / 2 + 1

Answer

g(x) = 2x - 1

Answer

g(x) = 1 - 2x

Question 36

Evaluate (f o g)(x) if f(x) = x^2 and g(x) = x + 2:

Accepted Answer

x^2 + 4x + 4

Answer

2x^2 + 4

Answer

x^4 + 4

Answer

x^2 + 2x + 4

Question 37

What is the domain of the function f(x) = sqrt(x + 2)?

Accepted Answer

x ≥ -2

Answer

x > 0

Answer

-2 ≤ x ≤ 0

Answer

All real numbers

Question 38

Which of the following is the graph of the function f(x) = |x - 3|?

Accepted Answer

V-shaped graph with vertex at (3, 0)

Answer

Straight line with slope 1

Answer

Circle centered at (3, 0)

Answer

Parabola opening upwards

Question 39

Which of the following is the equation of the asymptote to the graph of the function f(x) = (x + 2) / (x - 1)?

Accepted Answer

x = 1

Answer

x = -2

Answer

y = -1

Answer

y = 1

Question 40

Which of the following is NOT a characteristic of a function?

Accepted Answer

The output can depend on multiple inputs

Answer

The input is related to the output by a rule

Answer

The function assigns one output to each input

Answer

The function can be represented graphically

Question 41

Find the inverse of the function f(x) = 2x + 1

Accepted Answer

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

Answer

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

Answer

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

Question 42

Find the equation of the line that passes through the points (1, 3) and (3, 7).

Accepted Answer

f(x) = 2x + 1

Answer

f(x) = 2x - 1

Answer

f(x) = x - 2

Answer

f(x) = x + 2

Question 43

Identify the quadratic function:

Accepted Answer

f(x) = x^2 - 4x + 3

Answer

g(x) = sqrt(x) + 2

Answer

h(x) = e^x - 1

Answer

j(x) = sin(x) + cos(x)

Question 44

Which of the following represents the composite function g(f(x))?

Accepted Answer

The function that takes the output of f(x) and plugs it into g(x)

Answer

The function that takes the output of g(x) and adds it to the input of f(x)

Answer

The function that takes the output of f(x) and subtracts it from the input of g(x)

Answer

The function that takes the input of g(x) and plugs it into f(x)

Question 45

Which of the following functions has a constant rate of change?

Accepted Answer

f(x) = 2x + 1

Answer

h(x) = e^x

Answer

g(x) = x^2 - 4

Answer

j(x) = sin(x)

Question 46

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

Accepted Answer

y = -x + 4

Answer

y = -x + 2

Answer

y = x + 4

Answer

y = x - 2