Q: Find the inverse of the function $$f(x) = \\frac{3x-1}{5x+2}.$$

$$f^{-1}(x$$) = \\frac{1+2x}{3-5x}$$

Q: Determine the partial fraction decomposition of $$\\frac{x+1}{x(x+2)}.$$

$$\\frac{1}{x} - \\frac{1}{x+2}$$

Q: Write the following as a single logarithm: $$4\\log_2$(x+1) - 2\\log_2$(x-1) - \\frac{1}{2}\\log_2$(x^2+1$$)$.

$$\\log_2$\\left(\\frac{(x+1)^4$}{(x-1)^2$ \\sqrt{x^2$+1}}\\right)$$

Question 1

Given $$f(x) = 25 - x^2$ and g(x$$) = $$x^2 - 4$, what is (f$$ \circ g)(-1)$$?

Accepted Answer

$$3$$

Question 2

Using the Rational Zero Theorem, which of the following is a possible rational zero of $$f(x) = 2x^4$ - 3x^2$ - 3x - 6$$?

Accepted Answer

$$\frac{3}{2}$$

Question 3

Solve analytically for the exact real solution of $$\ln\left(\frac{x-2}{x}\right) = 1.$$

Accepted Answer

$$x = \frac{2}{1-e}$$

Question 4

What is the domain of $$f(x) = \log_5$(x-1) - 3$$?

Accepted Answer

$$(1, \infty)$$

Question 5

Given the parabola $$(y-3)^2 = 12(x-2$$)$$, what is the equation of its directrix?

Accepted Answer

$$x = 1$$

Question 6

If a section of seats in an arena is arranged arithmetically such that the first row has 60 seats and each subsequent row has 3 more seats than the row in front of it, and there are 45 rows, how many seats are in the section?

Accepted Answer

$$6,435$$

Question 7

Find the inverse of the function $$f(x) = \frac{3x-1}{5x+2}.$$

Accepted Answer

$$f^{-1}(x$$) = \frac{1+2x}{3-5x}$$

Question 8

What is the sum of the infinite geometric series $$\sum_{k=0}^{\infty} 5(0.4)^k$$?

Accepted Answer

$$8.33$$

Question 9

Given $$R(x) = \frac{x^2$ + 3x + 2}{x-1}$$, what is the equation of the slant asymptote?

Accepted Answer

$$y = x + 1$$

Question 10

Completely simplify the difference quotient for $$f(x) = \frac{1}{x-1}$$: $$\frac{f(x+h)-f(x)}{h}.$$

Accepted Answer

$$\frac{-1}{(x-1)(x+h-1)}$$

Question 11

Determine the partial fraction decomposition of $$\frac{x+1}{x(x+2)}.$$

Accepted Answer

$$\frac{1}{x} - \frac{1}{x+2}$$

Question 12

Write the following as a single logarithm: $$4\log_2$(x+1) - 2\log_2$(x-1) - \frac{1}{2}\log_2$(x^2+1$$)$.

Accepted Answer

$$\log_2$\left(\frac{(x+1)^4$}{(x-1)^2$ \sqrt{x^2$+1}}\right)$$

Question 13

If $$f(x) = 5x^7$ - 7x - 1$$, what is $$f(-x)$$?

Accepted Answer

$$-5x^7 + 7x - 1$

Question 14

Given the polynomial $$f(x) = (x-1)^2$(x-2)(x+5)$$, how many distinct real zeros does it have, and which zeros touch or cross the x-axis?

Accepted Answer

3 distinct real zeros; $$x=1$$ touches, $$x=2$$ and $$x=-5$$ cross

Question 15

Solve the polynomial inequality $$6x^4 - 4x$$ \leq 5x + 1$$, expressing the solution set with interval notation.

Accepted Answer

$$(-\infty, 1]$$

Question 16

A mosquito population follows the law of uninhibited growth with 140,000 initially and 170,000 after 12 hours. How many days will it take to reach 500,000? (Round to the nearest hundredth.)

Accepted Answer

1.85 days

Question 17

Given $$f(x) = \sqrt{x-2}$$, what is the domain of $$f$$?

Accepted Answer

$$[2, \infty)$$

Question 18

Which of the following transformations will result in the graph of $$f(x) = \sqrt{x}$$ being shifted right 3 units, reflected across the y-axis, and shifted up 5 units?

Accepted Answer

$$f(x) = \sqrt{-(x-3)} + 5$$

Comprehensive Study Guidance for College Algebra/Precalculus Final Exam

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