Question 1

Solve for $$x in $$the equation $$9 - 2(3 - x) = 5x.$$

Accepted Answer

$$x = \frac{15}{7}$$

Question 2

Which of the following is the solution to the quadratic equation $$x^2 - 5x + 6 = 0$?

Accepted Answer

$$x = 2$$ and $$x = 3$$

Question 3

Given $$f(x) = x^2$ + 2x + 1$$, find $$f(x+h) - f(x)$$ and simplify.

Accepted Answer

$$2xh + h^2$ + 2h$$

Question 4

Solve the equation by completing the square: $$x^2 - 6x + 5 = 0$.

Accepted Answer

$$x = 1$$ and $$x = 5$$

Question 5

Which of the following is the correct solution set for the compound inequality $$8 \u003c 4x \u003c 16$$?

Accepted Answer

$$x \in (2, 4)$$

Question 6

Solve the absolute value inequality $$|x - 1| \u003c 2$$ and express the solution in interval notation.

Accepted Answer

$$(-1, 3)$$

Question 7

Find the equation of the line passing through $$(3,4)$$ and perpendicular to the line $$y = 2x.$$

Accepted Answer

$$y = -\frac{1}{2}x + \frac{11}{2}$$

Question 8

Given $$f(x) = x^2$ and g(x$$) = $$-x^2 + 7$, which graph represents the transformation of f(x$$)$$ to g(x$$)$$?

Accepted Answer

Reflect over x-axis, shift 7 units up

Question 9

Find the domain of the function $$f(x) = \frac{1}{x^2$ - 6}.$$

Accepted Answer

$$x 
eq \pm \sqrt{6}$$

Question 10

Given $$f(x) = (x+2)^2$ + 2$$, what is the vertex of the parabola?

Accepted Answer

$$(-2, 2)$$

Question 11

Solve the exponential equation $$e^{2x} = 8$ for x$.

Accepted Answer

$$x = \frac{1}{2} \ln 8$$

Question 12

Which of the following is the correct solution to the logarithmic equation $$\log(x^2$ - 4x) = 2$$?

Accepted Answer

$$x = 5$$ and $$x = -1$$

Question 13

A population is growing at a rate of 2.5% per year. If the current population is 17,638,000, what will the population be in 5 years? Use the formula $$P = P_0$(1 + r)^t.$$

Accepted Answer

$$P \approx 19,999,600$$

Question 14

Solve the rational inequality $$\frac{x-4}{x-2} \geq 0$$ and express the solution in interval notation.

Accepted Answer

$$(-\infty, 2) \cup [4, \infty)$$

Question 15

Find the zeros and their multiplicities for $$f(x) = (x-1)^2$(x+3)$$ and state whether the graph crosses or touches the x-axis at each zero.

Accepted Answer

Zero at $$x=1 ($$multiplicity 2, touches), zero at $$x=-3 ($$multiplicity 1, crosses)

Question 16

Which graph best represents the polynomial $$f(x) = -2x^6$ + 3x^3$(x+1)$$?

Accepted Answer

Graph with end behavior down on both sides, crosses at $$x=0$$ and $$x=-1$$, touches at $$x=3$$

Question 17

Solve the system of inequalities and graph the solution set: $$y \u003e 2x - 4$$ and $$y \u003c -x + 3.$$

Accepted Answer

Region above $$y = 2x - 4$$ and below $$y = -x + 3$$

Question 18

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

Accepted Answer

$$(x+1)^2$$

Question 19

Given $$f(x) = \sqrt{x+6}$$, what is the domain of $$f^{-1}(x$$)$$?

Accepted Answer

$$[0, \infty)$$

College Algebra Final Exam Review: Comprehensive Study Notes

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