Q: Find the domain, range, asymptotes, and intercepts of the function $$y = \frac{1}{(x+4)^2$} - 3.$$

Domain: $$x \neq -4$$, Range: $$(-3, \infty)$$, Vertical asymptote: $$x = -4$$, Horizontal asymptote: $$y = -3$$, Intercept: $$(0, -\frac{11}{16})$$

Q: Find the inverse of $$g(x) = \frac{x+2}{x-5}$$ and state the domain of $$g^{-1}(x$$)$$.

$$g^{-1}(x$$) = \frac{5x+2}{x-1}$$, Domain: x$$ \neq 1$$

Question 1

Which of the following is the correct factorization of $$x^2 - 9x$?

Accepted Answer

$$x(x-9)$$

Question 2

Given the quadratic equation $$x^2 + 8x = 20$, which method would NOT be appropriate for solving it?

Accepted Answer

Graphing only

Question 3

Find the center and radius of the circle: $$(x-3)^2$$ + $$y^2 = 64$.

Accepted Answer

Center: $$(3,0)$$, Radius: $$8$$

Question 4

What is the equation of the circle with center $$(1,-5)$$ that passes through the point $$(3,4)$$?

Accepted Answer

$$(x-1)^2$$ + $$(y+5)^2 = 85$

Question 5

A ball is thrown upwards. Its height in feet is given by $$h(t) = -16t^2$ + 80t + 30. $$How long does it take for the ball to reach its maximum height?

Accepted Answer

$$t = 2.5$$ seconds

Question 6

Find the domain and range of the function $$f(x) = x^2$ - 6x - 12.$$

Accepted Answer

Domain: $$(-\infty, \infty)$$, Range: $$[-16, \infty)$$

Question 7

Which of the following is the correct difference quotient for $$f(x) = \frac{2}{x}$$?

Accepted Answer

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

Question 8

Solve the rational equation: $$\frac{3}{x-4} = \frac{2}{x+4}.$$

Accepted Answer

$$x = -5$$

Question 9

Find the domain, range, asymptotes, and intercepts of the function $$y = \frac{1}{(x+4)^2$} - 3.$$

Accepted Answer

Domain: $$x 
eq -4$$, Range: $$(-3, \infty)$$, Vertical asymptote: $$x = -4$$, Horizontal asymptote: $$y = -3$$, Intercept: $$(0, -\frac{11}{16})$$

Question 10

Does the function $$y = x^2$ + 7x - 2$$ have an inverse? Why?

Accepted Answer

No, because it is not one-to-one over $$(-\infty, \infty).$$

Question 11

Find the inverse of the function $$f(x) = (x-2)^3$ + 7.$$

Accepted Answer

$$f^{-1}(x$$) = \sqrt[3]{x-7} + 2$$

Question 12

Find the domain, range, and equation of the asymptote for $$y = 2$$^{x+1}$$ + 5.$$

Accepted Answer

Domain: $$(-\infty, \infty)$$, Range: $$(5, \infty)$$, Asymptote: $$y = 5$$

Question 13

Find the $$x-$$intercept and $$y-$$intercept of the function $$f(x) = 5$$^{x+1}$$ - 1.$$

Accepted Answer

$$x-$$intercept: $$-2.0$$, $$y-$$intercept: $$(0, 24)$$

Question 14

Find the equation of the exponential function $$y = a^x$$ that passes through the point $$(-3, 1).$$

Accepted Answer

$$y = 2^x$$

Question 15

Solve the equation $$5^{2x-1}$$ = \sqrt{5}$$ by relating the bases.

Accepted Answer

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

Question 16

Evaluate the function $$f(x) = 5^x at$$ $$x = -1, 0, \frac{1}{2}, 1, 2. $$Which is the correct set of values?

Accepted Answer

$$f(-1) = \frac{1}{5}, f(0) = 1, f(\frac{1}{2}) = \sqrt{5}, f(1) = 5, f(2) = 25$$

Question 17

Which of the following describes the vertical asymptote of $$f(x) = \frac{2x+1}{x-3}$$?

Accepted Answer

$$x = 3$$

Question 18

Find the inverse of $$g(x) = \frac{x+2}{x-5}$$ and state the domain of $$g^{-1}(x$$)$$.

Accepted Answer

$$g^{-1}(x$$) = \frac{5x+2}{x-1}$$, Domain: x$$ 
eq 1$$

Precalculus Exam Review: Factoring, Quadratics, Circles, Rational and Exponential Functions

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