Functions: Definition, Types, and Applications

Question 1

Which of the following correctly describes the domain and range of the exponential function f(x) = 2^x?

Accepted Answer

Domain: All real numbers, Range: All positive real numbers

Answer

Domain: All positive real numbers, Range: All real numbers

Answer

Domain: All real numbers, Range: All negative real numbers

Answer

Domain: All negative real numbers, Range: All positive real numbers

Question 2

How does changing the base of the exponent affect the graphs of exponential functions? Consider f(x) = 2^x and g(x) = 3^x.

Accepted Answer

The graphs will stretch or compress horizontally.

Answer

The graphs will shift vertically.

Answer

The graphs will have different y-intercepts.

Question 3

Which of the following is NOT a common real-world application of exponential functions?

Accepted Answer

Predicting the weather

Answer

Predicting the spread of a virus

Answer

Describing the growth of a savings account with compound interest

Answer

Modeling radioactive decay

Question 4

A city's population is currently 50,000 and is increasing by 2,500 people each year. Which function accurately models the population growth?

Accepted Answer

f(x) = 2500x + 50000

Answer

f(x) = 50000x + 2500

Answer

f(x) = 2500x - 50000

Answer

f(x) = 50000 - 2500x

Question 5

Which of the following is not a defining characteristic of a function?

Accepted Answer

A graph that resembles a parabola

Answer

A unique output for each input

Answer

A set of ordered pairs

Answer

The output depends on the input

Question 6

Identify the equation that represents a linear function:

Accepted Answer

y = 2x + 3

Answer

y = x^2 - 4

Answer

y = e^(2x)

Answer

y = |x| - 2

Question 7

Determine the domain of the function f(x) = √(x - 5):

Accepted Answer

[5, ∞)

Answer

[0, ∞)

Answer

(0, ∞)

Answer

(5, ∞)

Question 8

What is the vertex of the parabola represented by the equation y = (x - 2)^2 + 1?

Accepted Answer

(2, 1)

Answer

(0, 1)

Answer

(2, 5)

Answer

(0, 5)

Question 9

Which of the following is the graph of an exponential function?

Accepted Answer

An exponential curve (increasing)

Answer

A straight line

Answer

A parabola

Answer

A sine curve

Question 10

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

Accepted Answer

[1, ∞)

Answer

[0, ∞)

Answer

(-∞, ∞)

Answer

(-∞, 1]

Question 11

Which of the following transformations would translate the graph of y = x^2 left 3 units?

Accepted Answer

y = (x - 3)^2

Answer

y = (x + 3)^2

Answer

y = x^2 - 3

Answer

y = x^2 + 3

Question 12

What is the inverse of the function f(x) = 2x + 5?

Accepted Answer

f^-1(x) = x/2 - 5/2

Answer

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

Answer

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

Answer

f^-1(x) does not exist

Question 13

What is the rate of change of the function f(x) = 3x - 2?

Accepted Answer

3

Answer

1

Answer

2

Answer

4

Question 14

Which of the following is the graph of a linear function?

Accepted Answer

A straight line

Answer

A parabola

Answer

A circle

Answer

An ellipse

Question 15

What is the slope of the line that passes through the points (2, 3) and (4, 7)?

Accepted Answer

2

Answer

1

Answer

3

Answer

4

Question 16

Which of the following is the inverse of the function f(x) = x + 5?

Accepted Answer

f^-1(x) = x - 5

Answer

f^-1(x) = 5 - x

Answer

f^-1(x) = x/5

Answer

f^-1(x) = 5x

Question 17

Which function is increasing for all real numbers in the domain (-∞, ∞)?

Accepted Answer

f(x) = e^x

Answer

f(x) = x^3

Answer

f(x) = -x^2

Answer

f(x) = sin(x)

Question 18

Which of the following equations represents a linear function?

Accepted Answer

y = mx + b

Answer

y = x^2 + c

Answer

y = e^x + d

Answer

y = ln(x) + f

Question 19

Which of the following equations represents an exponential function?

Accepted Answer

y = e^x + d

Answer

y = mx + b

Answer

y = x^2 + c

Answer

y = ln(x) + f

Question 20

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

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 21

What is the equation of the exponential function with initial value 3 and growth rate 2?

Accepted Answer

f(x) = 3(2)^x

Answer

f(x) = 3x + 2

Answer

f(x) = 3x² + 2

Answer

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

Question 22

The graph of which type of function is a parabola?

Accepted Answer

Quadratic

Answer

Linear

Answer

Exponential

Answer

Rational

Question 23

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

Accepted Answer

x ≥ 2

Answer

x > 2

Answer

x ≤ 2

Answer

All real numbers

Question 24

Find the slope of the line passing through the points (2, 5) and (4, 9).

Accepted Answer

3

Answer

1

Answer

2

Answer

4

Question 25

How are a function and its inverse related?

Accepted Answer

Their graphs are reflections of each other over the line y = x

Answer

They share the same graph

Answer

They have the same domain and range

Answer

They have no relationship

Question 26

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

Accepted Answer

Inputs can be repeated

Answer

Every input has only one output

Answer

The domain is the set of all possible inputs

Answer

The range is the set of all possible outputs

Question 27

Which of the following is the graph of a linear function?

Accepted Answer

A straight line.

Answer

A parabola.

Answer

A circle.

Answer

A hyperbola.

Question 28

Which of these is an example of an exponential function?

Accepted Answer

f(x) = 2^x

Answer

f(x) = 2x

Answer

f(x) = x^2

Answer

f(x) = sin(x)

Question 29

Determine the set of values that x can take in the function f(x) = 1/x.

Accepted Answer

All real numbers except 0.

Answer

All positive numbers.

Answer

All negative numbers.

Answer

The set {x | x > 1}

Question 30

Analyze the function f(x) = sin(x) + cos(x). Determine its period and amplitude, and explain how these characteristics influence the graph of the function.

Accepted Answer

Period: 2π; Amplitude: √2

Answer

Period: π; Amplitude: 1

Answer

Period: √2; Amplitude: 2π

Question 31

A radioactive substance decays exponentially. After 2 hours, half of the substance has decayed. What is the half-life of the substance? How does this relate to the decay rate constant?

Accepted Answer

The half-life is 1 hour and is inversely proportional to the decay rate constant.

Answer

The half-life is 2 hours and is directly proportional to the decay rate constant.

Answer

The half-life is 0.693 hours and is not related to the decay rate constant.

Answer

It cannot be determined from the information given.

Question 32

Which of the following is not a defining characteristic of a function?

Accepted Answer

Output values can be repeated.

Answer

Each input has exactly one output.

Answer

Input values are independent.

Answer

The function can be represented by a graph.

Question 33

Which of the following functions is even?

Accepted Answer

f(x) = |x|

Answer

f(x) = x³

Answer

f(x) = sin(x)

Answer

f(x) = e^x

Question 34

Which of the following is the graph of a linear function?

Accepted Answer

A straight line

Answer

A parabola

Answer

A circle

Answer

A hyperbola

Question 35

Which of the following is the equation of the line with a slope of -2 and a y-intercept of 3?

Accepted Answer

y = -2x + 3

Answer

y = 3x - 2

Answer

y = 2x - 3

Answer

y = 3x + 2

Question 36

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

Accepted Answer

All real numbers except 2

Answer

All real numbers

Answer

All positive real numbers

Answer

All negative real numbers

Question 37

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

Accepted Answer

All non-negative real numbers

Answer

All real numbers

Answer

All positive real numbers

Answer

All negative real numbers

Question 38

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 + 1) / 2

Answer

f⁻¹(x) = 2x - 1

Answer

f⁻¹(x) = 1 - 2x

Question 39

Identify the graph of a linear function.

Accepted Answer

A straight line

Answer

A parabola

Answer

A circle

Answer

An ellipse

Question 40

Which of the following equations represents a quadratic function?

Accepted Answer

y = x^2 - 4

Answer

y = x + 2

Answer

y = e^x

Answer

y = sin(x)

Question 41

Which of the following functions is an exponential function?

Accepted Answer

f(x) = e^x

Answer

f(x) = 2x

Answer

f(x) = x^2

Answer

f(x) = log(x)

Question 42

What is the range of the function f(x) = x^2?

Accepted Answer

All non-negative real numbers

Answer

All real numbers

Answer

All positive real numbers

Answer

All negative real numbers

Question 43

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

Accepted Answer

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

Answer

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

Answer

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

Answer

f^-1(x) does not exist

Question 44

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

Accepted Answer

The graph of the function slopes down from left to right.

Answer

The output values get larger as the input values increase.

Answer

The function has a minimum value.

Answer

The function is symmetric about the y-axis.

Question 45

Which of the following is NOT a transformation of a quadratic function?

Accepted Answer

Multiplication by a linear function

Answer

Vertical translation

Answer

Horizontal translation

Answer

Reflection over the x-axis

Question 46

Which of the following is the graph of a linear function?

Accepted Answer

A straight line

Answer

A circle

Answer

A parabola

Answer

A hyperbola

Question 47

Which of the following is a quadratic function?

Accepted Answer

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

Answer

f(x) = e^x

Answer

f(x) = sin(x)

Answer

f(x) = log(x)

Question 48

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

Accepted Answer

The same output can correspond to multiple inputs

Answer

The domain is a set of numbers

Answer

The graph of a function is a line or curve

Answer

Each input corresponds to exactly one output

Question 49

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

Accepted Answer

y = 2x + 3

Answer

y = sin(x)

Answer

y = 3/x

Answer

y = x^2 + 2

Question 50

What is the slope of the line represented by the equation y = 3x - 4?

Accepted Answer

3

Answer

-4

Answer

-1/4

Answer

1

Question 51

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) = -2x - 1

Answer

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

Answer

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

Question 52

What is the y-intercept of the exponential function y = 2^x - 3?

Accepted Answer

-3

Answer

0

Answer

2

Answer

1

Question 53

Which of the following is an exponential function?

Accepted Answer

f(x) = 2^x

Answer

f(x) = 1/x

Answer

f(x) = sin(x)

Answer

f(x) = x^3

Question 54

Which of the following is the inverse of the function f(x) = x^3 + 1?

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 55

Which of the following functions is periodic?

Accepted Answer

f(x) = sin(x)

Answer

f(x) = x^2

Answer

f(x) = e^x

Question 56

A relation that assigns exactly one output to each input is called a:

Accepted Answer

Function

Answer

Sequence

Answer

Graph

Answer

Table

Question 57

Which of the following graphs represents an exponential function with a base greater than 1?

Accepted Answer

An upward-sloping curve that levels off

Answer

A straight line with a positive slope

Answer

A downward-sloping curve that levels off

Answer

A straight line with a negative slope

Question 58

Which of the following is an example of a real-world application of a quadratic function?

Accepted Answer

Predicting the trajectory of a projectile launched vertically upwards

Answer

Predicting the weather patterns over a week

Answer

Estimating the population of a city which grows exponentially

Answer

Calculating the area of a circle

Question 59

Which of the following represents the composition of functions f(x) = x^2 - 1 and g(x) = 2x + 3?

Accepted Answer

f(g(x)) = 4x^2 + 12x + 8

Answer

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

Answer

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

Answer

g(f(x)) = 4x^2 + 12x + 8

Question 60

What is the vertex of the parabola y = 2x^2 - 8x + 3?

Accepted Answer

(2, -5)

Answer

(-2, 1)

Answer

(4, -9)

Answer

(0, 3)

Question 61

Which of the following functions is an exponential function?

Accepted Answer

y = 3^x

Answer

y = x^2

Answer

y = x + 3

Answer

y = sin(x)

Question 62

What is the domain of the function f(x) = sqrt(9 - x^2)?

Accepted Answer

-3 ≤ x ≤ 3

Answer

x > -3 or x < 3

Answer

-3 < x < 3

Answer

x < -3 or x > 3

Question 63

Which of the following is the inverse of the function f(x) = 3x - 2?

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 64

Which of the following is a real-world application of functions?

Accepted Answer

Modeling the growth of bacteria in a petri dish

Answer

Solving a system of equations

Answer

Finding the area of a circle

Answer

Simplifying a polynomial