$$\text{Area} = \int_a^b f(x) \, dx$$

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_2$ x^2$ + a_1$ x + a_0$, where n$ is a nonnegative integer and a_i$ are constants

Question 2

Given the function $$f(x) = \sqrt[4]{x}$$, which of the following best describes its domain?

Accepted Answer

All real numbers $$x \geq 0$$

Question 3

Which of the following is NOT an example of a rational function?

Accepted Answer

$$f(x) = \sqrt{x}$$

Question 4

If $$f(x) = x^3$ and g(x$$) = $$x^{1/3}$$, which statement is true about their graphs?

Accepted Answer

Both graphs are defined for all real $$x$$ and are symmetric about the origin

Question 5

Which transformation results in the graph of $$y = f(x) + c$$ from $$y = f(x)$$?

Accepted Answer

A vertical shift upward by $$c$$ units

Question 6

Consider the function $$f(x) = 2^x. $$Which of the following statements is true?

Accepted Answer

The function is defined for all real $$x$$ and is always positive

Question 7

Which of the following is the correct equivalence between logarithmic and exponential forms?

Accepted Answer

$$y = \log_b x is $$equivalent to $$b^y = x$$

Question 8

Suppose $$f(x) = \sin(x)$$ and $$g(x) = \sin(3x). $$How does the period of $$g(x)$$ compare to $$f(x)$$?

Accepted Answer

The period of $$g(x) is$$ $$\frac{2\pi}{3}$$, which is one-third the period of $$f(x)$$

Question 9

Which of the following is a power function but NOT a polynomial?

Accepted Answer

$$f(x) = x$$^{1/2}$$

Question 10

A function $$f(x) is $$reflected across the $$y-$$axis. Which of the following represents the new function?

Accepted Answer

$$f(-x)$$

Question 11

Which of the following describes the effect of multiplying a function $$f(x) by a $$constant $$a \u003e 1 to $$get $$g(x) = a f(x)$$?

Accepted Answer

Vertical stretch by a factor of $$a$$

Question 12

Given $$f(x) = x^4$ - 3x^2$ + 1$$, what is the degree of the polynomial?

Accepted Answer

4

Question 13

Which of the following is the correct domain for the function $$f(x) = \sqrt[3]{x}$$?

Accepted Answer

All real numbers $$x$$

Question 14

If $$f(x) = x^2$ and g(x) = f(x-3$$)$$, how is the graph of g(x$$)$$ related to f(x$$)$$?

Accepted Answer

It is shifted 3 units to the right

Question 15

Which of the following is true about the graph of $$y = -f(x)$$ compared to $$y = f(x)$$?

Accepted Answer

It is reflected across the $$x-$$axis

Question 16

Which of the following is a transcendental function?

Accepted Answer

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

Question 17

Suppose you want to find the area under the curve $$y = f(x)$$ from $$x = a to$$ $$x = b. $$Which formula would you use?

Accepted Answer

$$	ext{Area} = \int_a^b f(x) \, dx$$

Question 18

Which of the following is the correct transformation for $$y = f(ax)$$ where $$a \u003e 1$$?

Accepted Answer

Horizontal shrink by a factor of $$a$$

Question 19

A student claims that $$y = \sqrt[6]{x} is $$defined for all real $$x. Is $$this correct?

Accepted Answer

No, it is only defined for $$x \geq 0$$

Representing Functions in Calculus: Types, Properties, and Transformations

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