Q: Given $$P(x) = -2x(x-5)^2$(x+\sqrt{7})^3$, what is the zero of P(x$$)$$ with the highest multiplicity, and what is that multiplicity?

$$x = -\sqrt{7}, \, \text{multiplicity 3}$$

Q: Which of the following best describes the end behavior of $$f(x) = -x^4$ + 6x^3$ - 9x^2$?

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

Question 1

Find a polynomial function of degree 4 with real zeros at $$x = -3 ($$multiplicity 2), $$x = 7 ($$multiplicity 1), and $$x = \dfrac{1}{3} ($$multiplicity 1). Which of the following is such a polynomial (up to a nonzero constant multiple)?

Accepted Answer

$$f(x) = (x + 3)^2$ (x - 7) (3x - 1)$$

Question 2

Given $$P(x) = -2x(x-5)^2$(x+\sqrt{7})^3$, what is the zero of P(x$$)$$ with the highest multiplicity, and what is that multiplicity?

Accepted Answer

$$x = -\sqrt{7}, \, 	ext{multiplicity 3}$$

Question 3

Which of the following best describes the end behavior of $$f(x) = -x^4$ + 6x^3$ - 9x^2$?

Accepted Answer

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

Question 4

For the rational function $$Q(x) = \dfrac{2x^2$ + 6x - 20}{x^2$ + 8x + 15}$$, which of the following statements is true?

Accepted Answer

There is a vertical asymptote at $$x = -5$$ and a hole at $$x = -3$$

Question 5

Solve the inequality $$x^3$$ - $$x^2$$ \u003c 12x$$. Which of the following intervals is the solution?

Accepted Answer

$$(-\infty, -3) \cup (0, 4)$$

Question 6

Solve the rational inequality $$\dfrac{2x + 3}{x - 5} \geq 1. $$What is the solution set?

Accepted Answer

$$(-\infty, 1] \cup (5, \infty)$$

Question 7

Solve the system of equations using elimination:

$$\begin{cases} 2x - 3y = -3 \ 3x - 4y = -12 \end{cases}$$

What is the solution $$(x, y)$$?

Accepted Answer

$$(0, -1)$$

Question 8

Solve the system using substitution:

$$\begin{cases} -2x + 5y = -1 \ 3x - 6y = -12 \end{cases}$$

What is the solution $$(x, y)$$?

Accepted Answer

$$(2, 1)$$

Question 9

Let $$f(x) = \dfrac{1}{x}$$ and $$g(x) = \dfrac{1}{x-4}. $$What is $$f \circ g(x)$$ and its domain?

Accepted Answer

$$f \circ g(x) = \dfrac{1}{\frac{1}{x-4}} = x-4$$, domain: $$x 
eq 4$$

Question 10

Let $$s(x) = \sqrt{x + 5}$$ and $$h(x) = x^2$ + 3. $$What is $$h \circ s(x)$$ and its domain?

Accepted Answer

$$h \circ s(x) = (\sqrt{x + 5})^2$ + 3 = x + 8$$, domain: $$x \geq -5$$

Question 11

A spherical balloon has surface area $$S(r) = 4\pi r^2$ and its radius as a function of time is r(t$$) = \dfrac{2}{3} $$t^3$$, $$t \geq 0. $$What is the surface area as a function of time?

Accepted Answer

$$S(t) = 4\pi \left(\dfrac{2}{3} t^3$\$$right)^2$$ = \dfrac{16\pi}{9} t^6$

Question 12

Given the function $$f(x) = \dfrac{3x}{x+2}$$, what is its inverse $$f^{-1}(x$$)$$?

Accepted Answer

$$f^{-1}(x$$) = \dfrac{2x}{3-x}$$

Question 13

Which of the following verifies that $$f(x) = \dfrac{x+1}{x}$$ and $$g(x) = \dfrac{1}{x-1}$$ are inverse functions?

Accepted Answer

$$f(g(x)) = x$$ and $$g(f(x)) = x$$

Question 14

Suppose a rational function $$R(x) = \dfrac{x-1}{x^3$ - 6x^2$ + 9x}. $$Which of the following describes all its vertical asymptotes?

Accepted Answer

$$x = 0$$ and $$x = 3$$

Question 15

Given a polynomial $$P(x) of $$degree $$n$$, what is the maximum number of turning points its graph can have?

Accepted Answer

$$n-1$$

Question 16

If a polynomial $$f(x)$$ has a zero at $$x = 2$$ with multiplicity 3, what does the graph do at $$x = 2$$?

Accepted Answer

It crosses the $$x-$$axis and flattens out at $$x = 2$$

Question 17

Which of the following is the correct domain for the function $$f(x) = \dfrac{1}{x^2$ - 9}$$?

Accepted Answer

$$x 
eq 3, x 
eq -3$$

Question 18

If $$f(x) is a $$one-to-one function, which of the following statements is always true about its inverse $$f^{-1}(x$$)$$?

Accepted Answer

$$f(f^{-1}(x)) = x$ for all x$ in the domain of f$$^{-1}$$

Question 19

Which of the following best describes the horizontal asymptote of the rational function $$f(x) = \dfrac{3x^2$ + 2}{2x^2$ - 5}$$?

Accepted Answer

$$y = \dfrac{3}{2}$$

College Algebra Exam 3 Review – Step-by-Step Study Guidance

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