Q: Given $$f(x) = 5x^4$ - 9x^2$ - 19$$, which of the following is true about its end behavior?

As $$x \to \pm\infty$$, $$f(x) \to \infty$$

Question 1

Given the rational function $$f(x) = \frac{7x^2$ - 7x - 210}{x^2$ + 6x - 7}$$, which of the following is the correct set of equations for its vertical and horizontal asymptotes?

Accepted Answer

$$x = -7, x = 1$$; \ y = 7$

Question 2

Use the Intermediate Value Theorem to determine if the polynomial $$f(x) = 5x^4$ - 9x^2$ - 19$$ has a real zero between $$x = -2$$ and $$x = 1.$$

Accepted Answer

Yes

Question 3

For the function $$g(x) = e^x + 15$$, what is the domain, range, and horizontal asymptote?

Accepted Answer

Domain: $$(-\infty, \infty)$$; Range: $$(15, \infty)$$; Horizontal asymptote: $$y = 15$$

Question 4

Which graph correctly represents the function $$f(x) = \log_{1/4}(x + 5) - 4$$?

Accepted Answer

Vertical asymptote at $$x = -5$$, graph shifted down by 4 units

Question 5

Solve for $$x in $$the equation $$M = R \left( \frac{a}{b} + 1 \right)^x$$ using logarithms.

Accepted Answer

$$x = \frac{\log(M/R)}{\log(a/b) + 1}$$

Question 6

Find the value of $$f(\log_3$(3 \ln 4))$$ for the function $$f(x) = 4^x.$$

Accepted Answer

$$3 \ln 4$$

Question 7

Which of the following describes the domain and range of the function $$f(x) = \log_{1/4}(x + 5) - 4$$?

Accepted Answer

Domain: $$(-5, \infty)$$; Range: $$(-\infty, \infty)$$

Question 8

Given $$f(x) = 5x^4$ - 9x^2$ - 19$$, which of the following is true about its end behavior?

Accepted Answer

As $$x 	o \pm\infty$$, $$f(x) 	o \infty$$

Question 9

For the rational function $$f(x) = \frac{7x^2$ - 7x - 210}{x^2$ + 6x - 7}$$, what are the $$x-$$intercepts?

Accepted Answer

$$x = 7, x = -3$$

Question 10

A population of bacteria grows according to $$P(t) = e$$^{t}$$ + 15. $$What is the minimum population size?

Accepted Answer

$$15$$

Question 11

If $$f(x) = \log_{1/4}(x + 5) - 4$$, what is the equation of its vertical asymptote?

Accepted Answer

$$x = -5$$

Question 12

Which transformation is applied to $$f(x) = e^x to $$obtain $$g(x) = e^x + 15$$?

Accepted Answer

Vertical shift upward by 15 units

Question 13

Given $$f(x) = 4^x$$, what is $$f(\log_4$ 16)$$?

Accepted Answer

$$16$$

Question 14

Which of the following is the correct inverse of $$f(x) = e^x + 15$$?

Accepted Answer

$$f^{-1}(x$$) = \ln(x - 15)$$

Question 15

If $$f(x) = \log_{1/4}(x + 5) - 4$$, what is the $$y-$$intercept?

Accepted Answer

$$f(-5)$$ $$ is undefined, so there is no y$-intercept

Question 16

For the equation $$M = R \left( \frac{a}{b} + 1 \right)^x$$, which step is necessary to isolate $$x$$?

Accepted Answer

Take the logarithm of both sides

Question 17

Which of the following is the correct range for $$g(x) = e^x + 15$$?

Accepted Answer

$$(15, \infty)$$

Question 18

If $$f(x) = 5x^4$ - 9x^2$ - 19$$, what is the degree and leading coefficient of the polynomial?

Accepted Answer

Degree: 4; Leading coefficient: 5

Key Concepts in College Algebra: Rational, Polynomial, Exponential, and Logarithmic Functions

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