Q: Given $$f(x) = x^2$ + 3x - 4$$ and $$g(x) = 5x - 3$$, what is $$(f+g)(x)$$ and its domain?

$$(f+g)(x) = x^2$ + 8x - 7$$, \text{ domain: } (-\infty, \infty)$

Q: Given $$f(x) = x^2$ + 3x - 4$$ and $$g(x) = 5x - 3$$, what is $$\left(\frac{f}{g}\right)(x)$$ and its domain?

$$\frac{x^2$ + 3x - 4}{5x - 3}$$, \text{ domain: } x \neq \frac{3}{5}$

Q: Given $$f(x) = \frac{1}{x}$$ and $$g(x) = \frac{4x}{x+1}$$, find $$(f \circ g)(x)$$ and its domain.

$$(f \circ g)(x) = \frac{x+1}{4x}$$, \text{ domain: } x \neq 0, x \neq -1$

Q: Express $$f(x) = x^2$ + 4x - 1 in $$the form $$f(x) = a(x-h)^2$ + k. $$What is the vertex?

$$f(x) = (x+2)^2$ - 5$$, \text{ vertex: } (-2, -5)$

Q: State the end behavior of the polynomial $$P(x) = -2x^4$ + 3x^2$ - 2x + 4.$$

As $$x \to \infty$$, $$P(x) \to -\infty$$; as $$x \to -\infty$$, $$P(x) \to -\infty$$

Question 1

Given the piecewise function

$$ f(x) = \begin{cases} 2x & 	ext{if } x  3 \end{cases} $$

What is the value of $$f(0)$$?

Accepted Answer

$$f(0) = 3$$

Question 2

Given $$f(x) = x^2$ + 3x - 4$$ and $$g(x) = 5x - 3$$, what is $$(f+g)(x)$$ and its domain?

Accepted Answer

$$(f+g)(x) = x^2$ + 8x - 7$$, 	ext{ domain: } (-\infty, \infty)$

Question 3

Given $$f(x) = x^2$ + 3x - 4$$ and $$g(x) = 5x - 3$$, what is $$\left(\frac{f}{g}\right)(x)$$ and its domain?

Accepted Answer

$$\frac{x^2$ + 3x - 4}{5x - 3}$$, 	ext{ domain: } x 
eq \frac{3}{5}$

Question 4

Given $$f(x) = \frac{1}{x}$$ and $$g(x) = \frac{4x}{x+1}$$, find $$(f \circ g)(x)$$ and its domain.

Accepted Answer

$$(f \circ g)(x) = \frac{x+1}{4x}$$, 	ext{ domain: } x 
eq 0, x 
eq -1$

Question 5

State the sequence of transformations that takes $$f(x) = x^2$ to g(x$$) = $$-2(x+4)^2 + 4$.

Accepted Answer

Shift left 4 units, vertical stretch by 2, reflect over x-axis, shift up 4 units

Question 6

Find the equation of the graph obtained by taking $$y = x^2$, stretching it vertically by a factor of 3, shifting up 3 units, and left 4 units.

Accepted Answer

$$y = 3(x+4)^2$ + 3$$

Question 7

Find $$(2 - 7i) + (8 + 4i).$$

Accepted Answer

$$10 - 3i$$

Question 8

Find $$(3 - 5i)(8 - 2i).$$

Accepted Answer

$$34 - 46i$$

Question 9

Solve the equation $$x^2 + 8x = 9$ by completing the square.

Accepted Answer

$$x = -4 \pm \sqrt{25}$$

Question 10

Solve the equation $$3x^2 + x = 5$ using the quadratic formula.

Accepted Answer

$$x = \frac{-1 \pm \sqrt{61}}{6}$$

Question 11

Express $$f(x) = x^2$ + 4x - 1 in $$the form $$f(x) = a(x-h)^2$ + k. $$What is the vertex?

Accepted Answer

$$f(x) = (x+2)^2$ - 5$$, 	ext{ vertex: } (-2, -5)$

Question 12

Solve the equation $$\frac{2}{x+5} + \frac{1}{x-5} = \frac{16}{x^2$-25}.$$

Accepted Answer

$$x = 3$$

Question 13

Solve the equation $$\sqrt{x+1} + 1 = x - 1.$$

Accepted Answer

$$x = 3$$

Question 14

Solve the inequality $$|2x + 5| \u003c 9. $$Express your answer in interval notation.

Accepted Answer

$$(-7, 2)$$

Question 15

State the end behavior of the polynomial $$P(x) = -2x^4$ + 3x^2$ - 2x + 4.$$

Accepted Answer

As $$x 	o \infty$$, $$P(x) 	o -\infty$$; as $$x 	o -\infty$$, $$P(x) 	o -\infty$$

Question 16

Solve the inequality $$\frac{4x+1}{x+2} \leq 3. $$Express your answer in interval notation.

Accepted Answer

$$(-\infty, -3] \cup (-2, 1]$$

Question 17

A company models its profit $$P(x) in $$thousands of dollars by $$P(x) = -2x^2$ + 8x + 10$$, where $$x is $$the number of units sold in hundreds. What is the maximum profit, and for how many units sold does it occur?

Accepted Answer

Maximum profit is $$26$$ thousand dollars at $$x = 2 (i.e$$., 200 units)

Question 18

Which of the following describes the domain of the function $$f(x) = \sqrt{5 - x}$$?

Accepted Answer

$$(-\infty, 5]$$

Question 19

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

Accepted Answer

$$9$$

College Algebra Review: Functions, Transformations, Equations, and Inequalities

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