Q: Which of the following best describes the end behavior of $$f(x) = -2x^4$?

As $$x \to \infty$$, $$f(x) \to -\infty$$; as $$x \to -\infty$$, $$f(x) \to -\infty.$$

Q: Find the inverse of $$f(x) = \dfrac{1}{x+3}$$ and state its domain.

$$f^{-1}(x$$) = \dfrac{1}{x} - 3$$, domain: x$$ \neq 0$$

Q: A $10,000 investment earns 5.5% annual interest, compounded continuously. What is the value after 5 years? (Round to the nearest dollar.)

$$A = 10,000 \cdot e$$^{0.055 \times 5}$$ \approx 13,077$$

Question 1

Which of the following is NOT a polynomial function?

Accepted Answer

$$f(x) = x^2$ + 3\sqrt{x}$$

Question 2

Given $$f(x) = (x-2)^3$(x-1)^2$$, which statement best describes the behavior of the graph at $$x=1$?

Accepted Answer

The graph touches the x-axis at $$x=1$$ but does not cross.

Question 3

Which of the following rational functions has a vertical asymptote at $$x = -4$$?

Accepted Answer

$$f(x) = \dfrac{x+8}{x^2$-64}$$

Question 4

Find the horizontal asymptote of $$f(x) = \dfrac{12x}{3x^2$ + 1}.$$

Accepted Answer

$$y = 0$$

Question 5

Which of the following best describes the end behavior of $$f(x) = -2x^4$?

Accepted Answer

As $$x 	o \infty$$, $$f(x) 	o -\infty$$; as $$x 	o -\infty$$, $$f(x) 	o -\infty.$$

Question 6

Let $$f(x) = x^2$ - 1$$ and $$g(x) = \sqrt{x+1}. $$What is the domain of the composite function $$(g \circ f)(x)$$?

Accepted Answer

$$\{x \mid x^2$ - 1 \geq -1\}$$

Question 7

Which of the following functions is one-to-one?

Accepted Answer

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

Question 8

Find the inverse of $$f(x) = \dfrac{1}{x+3}$$ and state its domain.

Accepted Answer

$$f^{-1}(x$$) = \dfrac{1}{x} - 3$$, domain: x$$ 
eq 0$$

Question 9

Rewrite the exponential equation $$3^{-4}$$ = \dfrac{1}{81}$$ as a logarithmic equation.

Accepted Answer

$$\log_3$ \dfrac{1}{81} = -4$$

Question 10

Which of the following is the correct simplification of $$3\log_a x - 2\log_a(x^2$ + x)$$?

Accepted Answer

$$\log_a \dfrac{x^3$}{(x^2$ + x)^2$}$$

Question 11

Solve for $$x$$: $$2\log_2$(x^2) + 3 = 11$.

Accepted Answer

$$x = 4$$

Question 12

A $10,000 investment earns 5.5% annual interest, compounded continuously. What is the value after 5 years? (Round to the nearest dollar.)

Accepted Answer

$$A = 10,000 \cdot e$$^{0.055 	imes 5}$$ \approx 13,077$$

Question 13

The population of a country grows from 6.04 million to 10 million in 50 years. Using $$A = A_0$ e$$^{kt}$$, what is the value of k$? (Round to 4 decimal places.)

Accepted Answer

$$k = \dfrac{\ln(10/6.04)}{50} \approx 0.0102$$

Question 14

A radioactive sample has a half-life of 5715 years. If you start with 16 grams, how much remains after 11,430 years?

Accepted Answer

$$2$$ grams

Question 15

A bottle of juice cools from $$70^\circF to$$ $$55^\circF in 10 $$minutes in a $$45^\circF $$refrigerator. Using Newton's Law of Cooling, what is the temperature after 15 minutes? (Round to the nearest degree.)

Accepted Answer

$$T \approx 50^\circF$$

Question 16

Which of the following is the correct application of the change of base formula for $$\log_2$ 7$$?

Accepted Answer

$$\log_2$ 7 = \dfrac{\log_{10} 7}{\log_{10} 2}$$

Question 17

Which of the following best describes the domain and range of $$f(x) = \log_3$ x$$?

Accepted Answer

Domain: $$(0, \infty)$$, Range: $$(-\infty, \infty)$$

Question 18

Solve for $$x$$: $$3 \cdot 2^x = 8.$$

Accepted Answer

$$x = 2$$

Polynomial and Rational Functions; Exponential and Logarithmic Functions: Study Guide

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