Q: Which of the following is the correct definition of a power function?

$$f(x) = x^a$$, \text{ where } a \text{ is a real number}$

Question 1

Which of the following is the correct definition of a polynomial function of degree $$n$$?

Accepted Answer

$$P(x) = a_n x^n + a_{n-1} x$$^{n-1}$$ + \cdots + a_1$ x + a_0$, where n$ is a nonnegative integer and a_n$$ 
eq 0$$

Question 2

Which of the following functions is NOT an algebraic function?

Accepted Answer

$$f(x) = \sin x$$

Question 3

Given the function $$f(x) = 2^x$$, which of the following statements is TRUE?

Accepted Answer

$$f(x) is an $$exponential function and is transcendental.

Question 4

Which of the following describes the effect of the transformation $$y = f(x) + c on $$the graph of $$y = f(x)$$, where $$c \u003e 0$$?

Accepted Answer

The graph shifts $$c$$ units upward.

Question 5

Suppose $$g(x) = f(x-3). $$How is the graph of $$g(x)$$ related to the graph of $$f(x)$$?

Accepted Answer

It is shifted 3 units to the right.

Question 6

Which of the following is the domain of the function $$f(x) = \sqrt[4]{x-2}$$?

Accepted Answer

$$[2, \infty)$$

Question 7

A function $$h(x) is $$defined as $$h(x) = \log_2$(x-1). $$What is the domain of $$h(x)$$?

Accepted Answer

$$(1, \infty)$$

Question 8

Which of the following is the correct equivalent exponential form of $$y = \log_b x$$?

Accepted Answer

$$b^y = x$$

Question 9

If $$f(x) = x^3$ and g(x) = 2f(x$$)$$, what transformation has been applied to f(x$$)$$ to obtain g(x$$)$$?

Accepted Answer

A vertical stretch by a factor of 2.

Question 10

Which of the following is a reflection of the graph of $$y = f(x)$$ across the $$x-$$axis?

Accepted Answer

$$y = -f(x)$$

Question 11

A medical researcher models the spread of a virus using $$N(t) = N_0$ e$$^{kt}$$, where N_0$ is the initial number of cases and k$$ \u003e 0$$. What type of function is N(t$$)$$?

Accepted Answer

An exponential function

Question 12

Which of the following is the correct period of the function $$f(x) = \sin(3x)$$?

Accepted Answer

$$\dfrac{2\pi}{3}$$

Question 13

Suppose $$f(x) = \sqrt[3]{x}$$ and $$g(x) = f(-x). $$Which of the following describes the transformation from $$f(x) to$$ $$g(x)$$?

Accepted Answer

Reflection across the $$y-$$axis

Question 14

Which of the following is a rational function?

Accepted Answer

$$f(x) = \dfrac{x+1}{x-2}$$

Question 15

A chemical reaction's rate is modeled by $$R(t) = 5 \log_{10}(t+1)$$, where $$t is $$time in seconds. What is the range of $$R(t)$$?

Accepted Answer

$$(-\infty, \infty)$$

Question 16

Which of the following best describes the effect of $$y = f(2x) on $$the graph of $$y = f(x)$$?

Accepted Answer

Horizontal shrink by a factor of $$\dfrac{1}{2}$$

Question 17

Which of the following is the correct area formula for a rectangle with base $$b$$ and height $$h$$?

Accepted Answer

$$A = b h$$

Question 18

If $$f(x) = x^2$ and g(x) = f(x+4$$)$$, what transformation has been applied to f(x$$)$$ to obtain g(x$$)$$?

Accepted Answer

Shift 4 units to the left

Question 19

Which of the following is the correct definition of a power function?

Accepted Answer

$$f(x) = x^a$$, 	ext{ where } a 	ext{ is a real number}$

Question 20

A population of bacteria doubles every hour. If the initial population is 100, which function models the population after $$t$$ hours?

Accepted Answer

$$P(t) = 100 \cdot 2^t$$

Representing Functions in Calculus: Types and Properties

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