Master Nonlinear Functions: Enhance Your Mathematical Modeling Skills

Question 1

Which of the following functions represents a linear relationship?

Accepted Answer

f(x) = 2x + 5

Answer

f(x) = sin(x)

Answer

f(x) = 2^x

Answer

f(x) = x^2

Question 2

The function f(x) = log(x - 2) has a vertical asymptote. What is the equation of this asymptote?

Accepted Answer

x = 2

Answer

x = -2

Answer

y = 2

Answer

y = -2

Question 3

A bacterial population doubles every hour. If the initial population is 100, which function accurately models the population growth after *t* hours?

Accepted Answer

P(t) = 100 * 2^t

Answer

P(t) = 100 * t

Answer

P(t) = 100 * t^2

Answer

P(t) = 100 + 2t

Question 4

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

Accepted Answer

π

Answer

2π

Answer

π/2

Answer

3π

Question 5

The function f(x) = e^x has a horizontal asymptote. What is the equation of this asymptote?

Accepted Answer

y = 0

Answer

x = 0

Answer

x = 1

Answer

y = 1

Question 6

A radioactive substance decays exponentially with a half-life of 5 years. If the initial amount is 10 grams, how much of the substance will remain after 15 years?

Accepted Answer

1.25 grams

Answer

5 grams

Answer

2.5 grams

Answer

0 grams

Question 7

Which of the following graphs represents the function f(x) = 1/x?

Accepted Answer

A graph with a hyperbola shape, with asymptotes at x = 0 and y = 0

Answer

A curve that approaches the x and y axes but never touches them

Answer

A parabola opening upwards

Answer

A straight line passing through the origin

Question 8

The function f(x) = sin(x) has a period of 2π. What is the period of the function g(x) = sin(2x - π/2)?

Accepted Answer

π

Answer

π/2

Answer

2π

Answer

4π

Question 9

Which of the following is NOT a type of nonlinear function?

Accepted Answer

Linear function

Answer

Trigonometric function

Answer

Exponential function

Answer

Logarithmic function

Question 10

What is the equation of the exponential function that has a graph that passes through the points (0, 1) and (1, e)?

Accepted Answer

y = e^x

Answer

y = e^(x-1)

Answer

y = x^e

Answer

y = 2^x

Question 11

The amplitude of the trigonometric function f(x) = 3cos(2x) is:

Accepted Answer

3

Answer

6

Answer

-3

Answer

2

Question 12

Which of the following is an application of nonlinear functions in mathematical modeling?

Accepted Answer

Modeling population growth

Answer

Solving systems of linear equations

Answer

Finding the slope of a line

Answer

Calculating the area of a circle

Question 13

The inverse of the exponential function is the:

Accepted Answer

Logarithmic function

Answer

Trigonometric function

Answer

Linear function

Answer

Quadratic function

Question 14

The domain of the function f(x) = sqrt(x-1) is:

Accepted Answer

[1, ∞)

Answer

[0, ∞)

Answer

(-∞, ∞)

Answer

(-∞, 1]

Question 15

The graph of the function f(x) = |x| is:

Accepted Answer

V-shaped

Answer

Parabolic

Answer

Linear

Answer

U-shaped

Question 16

Which of the following scenarios is best modeled by an exponential function?

Accepted Answer

The growth of bacteria in a petri dish, doubling every hour.

Answer

The height of a ball thrown vertically upwards.

Answer

The distance traveled by a car at a constant speed.

Question 17

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

Accepted Answer

(2, ∞)

Answer

(-∞, ∞)

Answer

[2, ∞)

Answer

(-∞, 2)

Question 18

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

Accepted Answer

2π

Answer

π

Answer

π/2

Answer

4π

Question 19

A population of rabbits is modeled by the equation P(t) = 100e^(0.2t), where t is time in months. What is the approximate population after 5 months?

Accepted Answer

271

Answer

150

Answer

1000

Answer

500

Question 20

The inverse function of f(x) = e^x is:

Accepted Answer

f⁻¹(x) = ln(x)

Answer

f⁻¹(x) = 1/e^x

Answer

f⁻¹(x) = e^(-x)

Answer

f⁻¹(x) = x/e

Question 21

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

Accepted Answer

3

Answer

6

Answer

1

Answer

2

Question 22

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

Accepted Answer

It has a vertical asymptote.

Answer

It has a horizontal asymptote.

Answer

It always passes through the origin.

Answer

It is always increasing.

Question 23

The equation of the tangent line to the graph of y = cos(x) at x = π/2 is:

Accepted Answer

y = -x + π/2

Answer

y = -x

Answer

y = x + π/2

Answer

y = x

Question 24

A radioactive substance decays according to the formula A(t) = A₀e^(-kt), where A(t) is the amount remaining after time t, A₀ is the initial amount, and k is the decay constant. If the half-life of the substance is 10 years, what is the value of k?

Accepted Answer

ln(2)/10

Answer

-ln(2)/10

Answer

ln(1/2)/10

Answer

10/ln(2)

Question 25

The graph of y = tan(x) has vertical asymptotes at:

Accepted Answer

x = (π/2) + nπ, where n is an integer

Answer

x = (π/3) + nπ, where n is an integer

Answer

x = (π/4) + nπ, where n is an integer

Answer

x = nπ, where n is an integer

Question 26

Which of the following scenarios is best modeled by an exponential function?

Accepted Answer

The growth of a bacterial colony, where the population doubles every hour.

Answer

The amount of water in a bathtub as it drains at a constant rate.

Answer

The distance traveled by a car moving at a constant speed.

Question 27

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

Accepted Answer

An exponential function with a base of 2.

Answer

A linear function with a slope of 2.

Answer

A logarithmic function with a base of 2.

Answer

A quadratic function with a leading coefficient of 2.

Question 28

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

Accepted Answer

(0, ∞)

Answer

[0, ∞)

Answer

(-∞, ∞)

Answer

(-∞, 0)

Question 29

Which of the following is equivalent to log(100)?

Accepted Answer

2

Answer

1

Answer

10

Answer

100

Question 30

The function f(x) = sin(x) is:

Accepted Answer

A periodic function with a period of 2π.

Answer

An exponential function with a base of e.

Answer

A quadratic function with a leading coefficient of 1.

Answer

A linear function with a slope of 1.

Question 31

Which of the following trigonometric identities is true?

Accepted Answer

sin^2(x) + cos^2(x) = 1

Answer

cos(2x) = 2cos(x)

Answer

tan(x) = sin(x) + cos(x)

Answer

sin(x + y) = sin(x) + sin(y)

Question 32

The graph of a function f(x) is symmetric about the y-axis. This implies that f(x) is:

Accepted Answer

An even function.

Answer

An odd function.

Answer

A periodic function.

Answer

A linear function.

Question 33

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

Accepted Answer

An odd function.

Answer

Neither even nor odd.

Answer

An even function.

Answer

A periodic function.

Question 34

The graph of a function f(x) is shifted 2 units to the right and 3 units up. The new function can be represented as:

Accepted Answer

f(x - 2) + 3

Answer

f(x - 2) - 3

Answer

f(x + 2) + 3

Answer

f(x + 2) - 3

Question 35

Which of the following graphs represents a function that is both increasing and concave down?

Accepted Answer

A graph that starts high, curves downwards, and continues to increase.

Answer

A graph that starts low, curves upwards, and continues to increase.

Answer

A graph that starts high, curves downwards, and decreases.