Q: Simplify the rational expression $$\dfrac{x^2$ - 25}{x^2$ + 6x + 5}$$ and state all restrictions.

$$\dfrac{x - 5}{x + 1}$$, $$x \neq -5, -1$$

Q: Which of the following best describes the end behavior of the polynomial $$f(x) = -3x^5$ + 2x^3$ - x + 7$$?

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

Q: Given $$f(x) = x^2$ - 9$$ and $$g(x) = x + 3$$, find $$(f/g)(x)$$ and state the domain.

$$\dfrac{x^2$ - 9}{x + 3}$$, $$x \neq -3$$

Question 1

A company models its revenue with the function $$R(t) = 500(1.06)^t$$, where $$t is $$the number of years since 2020. What is the revenue in 2023, and is this an example of exponential growth or decay?

Accepted Answer

$$R(3) = 595.51$$, exponential growth

Question 2

Given $$f(x) = x^2$ - 4x + 7$$, find $$f(-2).$$

Accepted Answer

$$f(-2) = 19$$

Question 3

What is the domain of the function $$f(x) = \sqrt{5 - 2x} in $$interval notation?

Accepted Answer

$$(-\infty, 2.5]$$

Question 4

If $$f(x) = 2x + 3$$ and $$g(x) = x^2$ - 1$$, what is $$(f \circ g)(x)$$?

Accepted Answer

$$2(x^2 - 1) + 3$

Question 5

Find the inverse of the function $$f(x) = 4x - 7.$$

Accepted Answer

$$f^{-1}(x$$) = \dfrac{x - 7}{4}$$

Question 6

A quadratic function $$f(x) = -2x^2$ + 8x - 3$$ models the height (in meters) of a ball thrown upward, where $$x is $$time in seconds. What is the maximum height reached by the ball?

Accepted Answer

$$f(2) = 5$$

Question 7

Factor the polynomial $$x^3$$ - $$2x^2 - 9x + 18$ completely.

Accepted Answer

$$(x - 2)(x - 3)(x + 3)$$

Question 8

Solve the inequality $$(x + 1)(x - 4) \leq 0$$ and express the solution in interval notation.

Accepted Answer

$$[-1, 4]$$

Question 9

Given $$f(x) = (x - 2)^2$(x + 5)$$, which statement about the graph is correct?

Accepted Answer

The graph crosses the x-axis at $$x = -5$$ and touches at $$x = 2.$$

Question 10

Simplify the rational expression $$\dfrac{x^2$ - 25}{x^2$ + 6x + 5}$$ and state all restrictions.

Accepted Answer

$$\dfrac{x - 5}{x + 1}$$, $$x 
eq -5, -1$$

Question 11

Solve the equation $$\sqrt{x + 2} = x - 2$$ and check for extraneous solutions.

Accepted Answer

$$x = 4 is $$the only solution

Question 12

A rational function is given by $$f(x) = \dfrac{x + 3}{x - 2}. $$Which of the following statements is true?

Accepted Answer

There is a vertical asymptote at $$x = 2$$ and a horizontal asymptote at $$y = 1.$$

Question 13

Solve the exponential equation $$5^x = 60$$ for $$x ($$round to three decimals).

Accepted Answer

$$x = \dfrac{\ln 60}{\ln 5} \approx 2.113$$

Question 14

Which of the following best describes the end behavior of the polynomial $$f(x) = -3x^5$ + 2x^3$ - x + 7$$?

Accepted Answer

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

Question 15

A population of bacteria triples every 4 hours. If the initial population is 200, what is the population after 8 hours?

Accepted Answer

$$200 	imes 3^2$ = 1800$$

Question 16

Given $$f(x) = x^2$ - 9$$ and $$g(x) = x + 3$$, find $$(f/g)(x)$$ and state the domain.

Accepted Answer

$$\dfrac{x^2$ - 9}{x + 3}$$, $$x 
eq -3$$

Question 17

Which of the following is the correct interpretation of $$\log_4$(64)$$?

Accepted Answer

$$4^3 = 64$,\ \log_4$(64) = 3$

Question 18

A quadratic model $$y = ax^2$ + bx + c is $$used to predict the height of a projectile. Why is it generally unreliable to use this model to predict the height at times far beyond the observed data range?

Accepted Answer

Extrapolation can lead to predictions that do not reflect real-world behavior.

College Algebra Study Guide: Functions, Polynomials, Rational, Radical, Exponential, and Logarithmic Topics

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