Question 1

Simplify the expression $$\frac{4x^2$ - 7x}{2x}.$$

Accepted Answer

$$2x - \frac{7}{2}$$

Question 2

Completely factor $$12x^2 - 27x$.

Accepted Answer

$$3x(4x - 9)$$

Question 3

For the rational function $$f(x) = \frac{x^2$ - 4}{x^2$ - x - 6}$$, find all vertical, horizontal, and oblique asymptotes.

Accepted Answer

Vertical: $$x = 3, x = -2$$; Horizontal: $$y = 1$$; No oblique asymptote.

Question 4

Simplify the expression $$\frac{2x^2$ - 8x}{x^2$ - 4}.$$

Accepted Answer

$$2 \frac{x-4}{x+2}$$

Question 5

Given $$f(x) = x^2$ - 5x + 6$$, find the equation of the line perpendicular to $$y = 2x - 5$$ and passing through the point $$(2, f(2)).$$

Accepted Answer

$$y = -\frac{1}{2}x + 5$$

Question 6

Express $$f(x) = 2x^2$ - 8x + 5 in $$the form $$a(x-h)^2 + k$.

Accepted Answer

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

Question 7

Solve $$4x^2 - 9 = 0$ and express the solution set in intervals.

Accepted Answer

$$x = \pm \frac{3}{2}$$

Question 8

Find the domain of $$f(x) = \sqrt{x-5}.$$

Accepted Answer

$$x \geq 5$$

Question 9

Find the range of the function $$f(x) = x^2$ - 4x + 7. $$Express your answer in interval notation.

Accepted Answer

$$[3, \infty)$$

Question 10

If the function $$f(x) = x^3$ is even, odd, or neither?

Accepted Answer

Odd

Question 11

Find the vertex and axis of symmetry of the parabola with vertex $$(2, -3)$$ passing through $$(0, 5).$$

Accepted Answer

Vertex: $$(2, -3)$$; Axis: $$x = 2$$

Question 12

Find the inverse of the function $$f(x) = 2x + 1.$$

Accepted Answer

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

Question 13

The synthetic division of $$x^3$$ - $$4x^2 + 5x - 2$ divided by x - 2$ yields which remainder?

Accepted Answer

-2

Question 14

If $$f(x) = x^3$ - 4x^2$ + 5x - 2$$, write $$f as a $$product of linear factors.

Accepted Answer

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

Question 15

Find all solutions of $$x^2 - 8x + 7 = 0$.

Accepted Answer

$$x = 1, x = 7$$

Question 16

Find the center and radius of the circle with equation $$x^2$$ + $$y^2 - 8x + 10y = 7$.

Accepted Answer

Center: $$(4, -5)$$; Radius: $$6$$

Question 17

Find the asymptotes for the function $$f(x) = \frac{3x^2$ + 2x - 5}{x^2$ - 8x + 10}.$$

Accepted Answer

Vertical: $$x = 5, x = 2$$; Horizontal: $$y = 3$$

Question 18

Given the price $$p (in $$dollars) and the quantity $$x$$ sold of a certain product obey the demand equation $$p = 100 - 2x$$, which model expresses the revenue $$R as a $$function of $$x$$?

Accepted Answer

$$R(x) = (100 - 2x)x$$

Comprehensive Precalculus Final Review: Key Concepts and Practice Problems

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