Functions: Master the Basics of Mathematical Relationships

Question 1

To shift the graph of f(x) = x^2 three units left and two units up, which transformation should be applied?

Accepted Answer

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

Answer

g(x) = (x - 3)^2 - 2

Answer

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

Question 2

Classify the function f(x) = sin(x) + cos(x):

Accepted Answer

Trigonometric

Answer

Exponential

Answer

Logarithmic

Answer

Linear

Question 3

How many relative extrema are there in the graph of the function f(x) = (x-2)²+1?

Accepted Answer

1

Answer

0

Answer

2

Question 4

A company's marketing budget is decreasing exponentially by 15% annually. If the current budget is $100,000, what will it be after 5 years?

Accepted Answer

$64,500

Answer

$75,000

Answer

$85,000

Answer

$50,000

Question 5

Which trigonometric identity can be used to simplify the expression cos^2(x) - sin^2(x)?

Accepted Answer

cos(2x)

Answer

sin(2x)

Answer

tan(2x)

Question 6

In cryptography, what is a potential application of the inverse of a function?

Accepted Answer

Decrypting encrypted messages

Answer

Creating random sequences

Answer

Compressing data efficiently

Question 7

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

Accepted Answer

Each input has exactly one output

Answer

The graph of a function is always a straight line

Answer

The domain and range of a function are always equal

Answer

Functions always have a positive slope

Question 8

Which of the following transformations shifts a graph vertically?

Accepted Answer

f(x) + c

Answer

f(x) - c

Answer

f(x + c)

Answer

f(x - c)

Question 9

Which of the following equations represents an exponential function?

Accepted Answer

y = 3^x

Answer

y = 2x

Answer

y = x^2

Answer

y = sin(x)

Question 10

What is the equation of the vertical asymptote of the function f(x) = 1/(x-2)?

Accepted Answer

x = 2

Answer

y = 0

Answer

y = 1

Answer

x = 0

Question 11

For the function f(x) = -x + 4, what is the y-intercept?

Accepted Answer

4

Answer

-4

Answer

0

Answer

8

Question 12

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

Accepted Answer

2π

Answer

π

Answer

π/2

Answer

3π

Question 13

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

Accepted Answer

The set of all non-negative real numbers

Answer

The set of all real numbers

Answer

The set of all positive real numbers

Answer

The set of all negative real numbers

Question 14

Which function has a range of 0 to 1?

Accepted Answer

f(x) = sin(x)

Answer

f(x) = x^2

Answer

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

Answer

f(x) = e^x

Question 15

Which transformation would shift the function f(x) = x^2 - 4 up by 2 units?

Accepted Answer

f(x) + 2

Answer

f(x - 2)

Answer

f(x) * 2

Answer

f(2x)

Question 16

Identify the function that models exponential growth:

Accepted Answer

y = 2^x + 3

Answer

y = 2x + 3

Answer

y = sin(x) + 3

Answer

y = log(x) + 3

Question 17

Identify the equation that accurately represents the logarithmic function depicted in the graph.

Accepted Answer

y = log(2, x)

Answer

y = 2^x

Answer

y = x^2

Answer

y = log(10, x)

Question 18

An exponential function f(x) = b^x has been transformed. Which transformation will shift the graph of f(x) vertically upwards by one unit?

Accepted Answer

f(x) → f(x) + 1

Answer

f(x) → 2f(x)

Answer

f(x) → f(x - 1)

Answer

f(x) → f(x^2)

Question 19

Determine the domain of the rational function: f(x) = (x - 2)/(x^2 - 4)

Accepted Answer

All real numbers except 0 and 2

Answer

All real numbers

Answer

Only real numbers greater than 0

Answer

Only real numbers between -2 and 2

Question 20

For the function f(x) = √(x - 3) + 2, identify its domain and range.

Accepted Answer

Domain: [3, ∞), Range: [2, ∞)

Answer

Domain: (-∞, 3], Range: [2, ∞)

Answer

Domain: (3, ∞), Range: (2, ∞)

Answer

Domain: (-∞, ∞), Range: (-∞, ∞)

Question 21

Which of the following is NOT a type of function?

Accepted Answer

Pattern Algebra

Answer

Exponential

Answer

Linear

Answer

Rational

Question 22

Which transformation is represented by the function y = sin(x + π/2)?

Accepted Answer

Horizontal shift right by π/2 units

Answer

Horizontal shift left by π/2 units

Answer

Vertical shift up by π/2 units

Answer

Vertical shift down by π/2 units

Question 23

Which of the following functions is an odd function?

Accepted Answer

f(x) = sin(x)

Answer

f(x) = x^2 - 1

Answer

f(x) = |x|

Answer

f(x) = x^3 + x

Question 24

Determine the domain and range of the function f(x) = x² + 1.

Accepted Answer

Domain: All real numbers, Range: [1, ∞)

Answer

Domain: All real numbers, Range: (-∞, ∞)

Answer

Domain: [0, ∞), Range: [1, ∞)

Answer

Domain: (-1, 1), Range: (0, 2)

Question 25

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

Accepted Answer

x > 2

Answer

x < 2

Answer

x ≥ 2

Answer

x ≤ 2

Question 26

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

Accepted Answer

h(x) = (x - 3)^2

Answer

h(x) = x^2 - 3

Answer

h(x) = x^2 + 3

Answer

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

Question 27

Which of the following is a characteristic of periodic functions?

Accepted Answer

They repeat their values at regular intervals.

Answer

Their graphs are always increasing.

Answer

They have a maximum and minimum value.

Answer

They are defined for all real numbers.

Question 28

What is the maximum value of the function f(x) = -x² + 4x?

Accepted Answer

4

Answer

1

Question 29

Which of the following relations represents a function?

Accepted Answer

f(x) = x^2

Answer

x = y^2

Answer

y^2 - x^2 = 0

Answer

x + y = 5

Question 30

Which graph corresponds to the function f(x) = x^2 - 4?

Accepted Answer

[Image of a parabola opening upwards with vertex at (0, -4)]

Answer

[Image of a parabola opening downwards with vertex at (0, -4)]

Answer

[Image of a line with y-intercept at -4 and slope 2]

Question 31

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) = 2x - 3

Question 32

Calculate the derivative of the function f(x) = e^x?

Accepted Answer

f'(x) = e^x

Answer

f'(x) = 1 / e^x

Answer

f'(x) = e^(x^2)

Answer

f'(x) = x e^x

Question 33

Which application utilizes exponential functions?

Accepted Answer

Modeling population growth

Answer

Calculating the area of a circle

Answer

Finding the volume of a cone

Answer

Measuring the angle of a triangle

Question 34

Which characteristic is NOT a property of a linear function?

Accepted Answer

The graph of a linear function is a circle.

Answer

The slope of a linear function is constant.

Answer

The graph of a linear function is a straight line.

Answer

The y-intercept of a linear function is the point where the graph crosses the y-axis.

Question 35

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

Accepted Answer

f(x) = 2^x

Answer

f(x) = sin(x) (trigonometric function)

Answer

f(x) = x + 2 (linear function)

Answer

f(x) = x^2 (quadratic function)

Question 36

Which of the following functions is continuous at x = 1?

Accepted Answer

f(x) = { x^2 if x ≠ 1, 2 if x = 1 }

Answer

f(x) = { 0 if x ≠ 1, 1 if x = 1 } (discontinuous at x = 1)

Answer

f(x) = { 1 if x ≠ 1, 0 if x = 1 } (discontinuous at x = 1)

Answer

f(x) = { x if x ≠ 1, 3 if x = 1 } (discontinuous at x = 1)

Question 37

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

Accepted Answer

The output can be related to multiple inputs

Answer

The input is related to a single output

Answer

The relation can be represented graphically

Question 38

Consider the following relation: {(1, 2), (2, 4), (3, 6), (4, 2)}. Is this relation a function?

Accepted Answer

No, the relation fails the vertical line test.

Answer

Yes, each input is paired with exactly one output.

Answer

Yes, the relation can be graphed as a straight line.

Answer

No, the relation is not linear.

Question 39

Which of the following graphs represents an exponential function?

Accepted Answer

A graph that increases rapidly as x increases

Answer

A graph that forms a circle (circular function)

Answer

A graph that decreases linearly as x increases (linear function)

Answer

A graph that is a straight line (linear function)

Question 40

Which of the following transformations shifts a trigonometric function horizontally to the left?

Accepted Answer

Adding a constant to x

Answer

Multiplying y by a constant (vertical transformation)

Answer

Adding a constant to y (vertical translation)

Answer

Multiplying x by a constant (horizontal stretch or compression)

Question 41

Consider the functions f(x) = 2x - 1 and g(x) = x². What is (f ∘ g)(x)?

Accepted Answer

4x² - 2x - 1

Answer

2x³ - 1 (incorrect composition)

Answer

x⁴ - 2x² + 1 (incorrect composition)

Answer

x² - 2x + 1 (incorrect composition)

Question 42

The function f(x) = 2x + 3 is a(n):

Accepted Answer

Linear function

Answer

Logarithmic function

Answer

Quadratic function

Answer

Exponential function

Question 43

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

Accepted Answer

2π

Answer

π

Answer

4π

Answer

π/2

Question 44

Which statement is NOT true about functions?

Accepted Answer

A function can have multiple outputs for the same input.

Answer

Every input in a function has a unique output.

Answer

Functions describe relationships between input and output.

Answer

Functions can be represented using graphs.

Question 45

Which type of function has a constant rate of change, meaning its graph is a straight line?

Accepted Answer

Linear

Answer

Trigonometric

Answer

Exponential

Answer

Logarithmic

Question 46

What is the domain of the function f(x) = √(x - 2)?  (In other words, what values of x can we use without getting an error?)

Accepted Answer

[2, ∞)

Answer

[-2, ∞)

Answer

(2, ∞)

Answer

[-∞, -2]

Question 47

Which of these transformations will shift a function's graph down?

Accepted Answer

f(x) - c

Answer

f(x - c)

Answer

f(x) + c

Answer

f(x + c)

Question 48

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

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 49

Which function repeats its pattern over a fixed interval, making it a periodic function?

Accepted Answer

f(x) = sin(x)

Answer

f(x) = e^x

Answer

f(x) = x^2

Answer

f(x) = √x

Question 50

What is the slope of the line that touches the curve of f(x) = x^3 at the point where x = 2?

Accepted Answer

12

Answer

24

Answer

6

Answer

3

Question 51

Which function has a highest point (maximum) at y = 5?

Accepted Answer

f(x) = -x^2 + 5

Answer

f(x) = -x^2 - 5

Answer

f(x) = x^2 - 5

Answer

f(x) = x^2 + 5

Question 52

What is the area of the shape formed between the graph of f(x) = 2x, the x-axis, and the vertical lines x = 0 and x = 2?

Accepted Answer

4

Answer

8

Answer

2

Question 53

Which real-world scenario is best represented by an exponential function?

Accepted Answer

The growth of a population over time

Answer

The motion of a pendulum swinging back and forth

Answer

The relationship between the force applied to a spring and its displacement

Answer

The path of a ball thrown in the air

Question 54

Which of the following is not an accepted representation of a function?

Accepted Answer

A set of ordered pairs with repeated first elements.

Answer

An equation.

Answer

A graph.

Question 55

The function f(x) = 3x + 1 is considered a(n):

Accepted Answer

Linear function.

Answer

Logarithmic function.

Answer

Exponential function.

Answer

Quadratic function.

Question 56

The graph of y = -f(x) is a reflection of the graph of y = f(x) across the:

Accepted Answer

x-axis.

Answer

line y = x.

Answer

y-axis.

Question 57

Given the functions f(x) = x^2 and g(x) = x - 1, find (f o g)(3).

Accepted Answer

8

Answer

4

Answer

6

Answer

9

Question 58

For the function f(x) = 2/(x + 1), what is its domain?

Accepted Answer

All real numbers except -1.

Answer

x ≥ -1.

Answer

All real numbers.

Answer

x ≤ -1.

Question 59

A population of bacteria doubles every hour. If the initial population is 200, what is the population after 4 hours?

Accepted Answer

3200

Answer

800

Answer

400

Answer

600

Question 60

The inverse function of f(x) = 5 - 2x is:

Accepted Answer

f⁻¹(x) = (5 - x)/2

Answer

f⁻¹(x) = -2x + 5

Answer

f⁻¹(x) = 2x - 5

Answer

f⁻¹(x) = 5 + 2x

Question 61

The graph of y = sin(x) exhibits a period of:

Accepted Answer

2π.

Answer

π/2.

Answer

4π.

Answer

π.

Question 62

Which characteristic is NOT applicable to a function?

Accepted Answer

Multiple outputs for a single input.

Answer

Well-defined domain and range.

Answer

Unique output for each input.

Question 63

The function f(x) = x^3 - 2x is:

Accepted Answer

A polynomial function.

Answer

A rational function.

Answer

An exponential function.

Answer

A logarithmic function.

Question 64

If the function *f* is defined by *f(x) = x² - 3x + 2*, what is the value of *f(4)*?

Accepted Answer

6

Answer

2

Answer

-2

Answer

10