Q: If $$f(x) = 2x + 3$$ and $$g(x) = x^2$, what is (f+g)(2$$)$$?

$$f(2) + g(2) = (2 \times 2 + 3) + (2^2$) = 7 + 4 = 11$$

Q: A company charges a flat fee of $$\50$ plus $$\$$20$$ per hour for consulting. Write a linear function $$C(h)$$ for the total cost for $$h$$ hours, and find the cost for $$5$$ hours.

$$C(h) = 20h + 50$$, so $$C(5) = 20 \times 5 + 50 = 150$$

Q: Given the piecewise function $$f(x) = \begin{cases} x+2 & x \u003c 0 \\ 2x-1 & x \geq 0 \end{cases}$$, what is $$f(-3) + f(2)$$?

$$(-3 + 2) + (2 \times 2 - 1) = (-1) + (4 - 1) = -1 + 3 = 2$$

Question 1

Given points $$A(2, 3)$$ and $$B(8, 11)$$, what is the distance between them?

Accepted Answer

$$\sqrt{(8-2)^2$ + (11-3)^2$} = \sqrt{36 + 64} = \sqrt{100} = 10$$

Question 2

What is the equation of the circle with center at $$(4, -2)$$ and passing through the point $$(7, 2)$$?

Accepted Answer

$$(x-4)^2$$ + $$(y+2)^2 = 25$

Question 3

Find the midpoint of the segment joining $$(5, -3)$$ and $$(-1, 7).$$

Accepted Answer

$$(2, 2)$$

Question 4

Given the function $$f(x) = \sqrt{9 - x}$$, what is the domain of $$f$$?

Accepted Answer

$$(-\infty, 9]$$

Question 5

If $$f(x) = 2x + 3$$ and $$g(x) = x^2$, what is (f+g)(2$$)$$?

Accepted Answer

$$f(2) + g(2) = (2 	imes 2 + 3) + (2^2$) = 7 + 4 = 11$$

Question 6

Which of the following graphs represents a function that passes the vertical line test?

Accepted Answer

A graph where every vertical line intersects the curve at most once.

Question 7

Given $$f(x) = x^2$ - 4x + 7$$, find the average rate of change from $$x = 1 to$$ $$x = 5.$$

Accepted Answer

$$\frac{f(5) - f(1)}{5-1} = \frac{(25 - 20 + 7) - (1 - 4 + 7)}{4} = \frac{12 - 4}{4} = 2$$

Question 8

Find the equation of the line passing through $$(2, -1)$$ and parallel to $$y = 3x + 5.$$

Accepted Answer

$$y = 3x - 7$$

Question 9

A company charges a flat fee of $$\50$ plus $$\$$20$$ per hour for consulting. Write a linear function $$C(h)$$ for the total cost for $$h$$ hours, and find the cost for $$5$$ hours.

Accepted Answer

$$C(h) = 20h + 50$$, so $$C(5) = 20 	imes 5 + 50 = 150$$

Question 10

Given $$f(x) = |x-3|$$, which of the following describes the transformation from $$f(x) = |x|$$?

Accepted Answer

A horizontal shift $$3$$ units to the right.

Question 11

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

Accepted Answer

$$f(g(2)) = f(3) = 9$$

Question 12

Given the piecewise function $$f(x) = \begin{cases} x+2 & x \u003c 0 \ 2x-1 & x \geq 0 \end{cases}$$, what is $$f(-3) + f(2)$$?

Accepted Answer

$$(-3 + 2) + (2 	imes 2 - 1) = (-1) + (4 - 1) = -1 + 3 = 2$$

Question 13

Find the zero of the linear function $$f(x) = 7 - 2x.$$

Accepted Answer

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

Question 14

Given $$f(x) = x^2$ - 1$$ and $$g(x) = \sqrt{x}$$, what is the domain of $$(g \circ f)(x)$$?

Accepted Answer

$$x^2 - 1$$ \geq 0 \implies x \leq -1$$ or x$$ \geq 1$$

Question 15

If $$f(x) = 3x - 4$$, find the difference quotient $$\frac{f(x+h) - f(x)}{h}.$$

Accepted Answer

$$\frac{3(x+h) - 4 - (3x - 4)}{h} = \frac{3h}{h} = 3$$

Question 16

Which of the following best describes the transformation from $$f(x) = x^2$ to g(x$$) = $$-2(x-1)^2 + 3$?

Accepted Answer

Reflect over the $$x-$$axis, vertically stretch by $$2$$, shift right $$1$$ unit, and up $$3$$ units.

Question 17

Suppose the graph of a function $$f is $$increasing on $$(-\infty, 2)$$, constant on $$[2, 5]$$, and decreasing on $$(5, \infty). $$Which interval contains a relative maximum?

Accepted Answer

$$x = 5$$

Question 18

Given $$f(x) = x^2$ and g(x) = 2x + 1$, find (f - g)(3$$)$$.

Accepted Answer

$$f(3) - g(3) = 9 - 7 = 2$$

Question 19

A biologist models the population of a bacteria culture by $$P(t) = 500 + 40t$$, where $$t is $$time in hours. After how many hours will the population reach $$900$$?

Accepted Answer

$$t = \frac{900 - 500}{40} = 10$$

College Algebra: Study Guide for Test 1 (Chapters 1 & 2)

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