Question 1

Given $$f(x) = \frac{2x^2$ - 3x + 1}{x}$$, what is $$f(-2)$$?

Accepted Answer

$$f(-2) = \frac{2(-2)^2$ - 3(-2) + 1}{-2} = \frac{8 + 6 + 1}{-2} = \frac{15}{-2} = -7.5$$

Question 2

Given $$f(x) = x^2$ - 4$$ and $$h is a $$real number, what is $$f(x+h) - f(x)$$?

Accepted Answer

$$f(x+h) - f(x) = (x+h)^2$ - 4 - (x^2$ - 4) = x^2$ + 2xh + h^2$ - 4 - x^2$ + 4 = 2xh + h^2$

Question 3

What is the domain of $$f(x) = \frac{1}{3x - 1}$$?

Accepted Answer

All real numbers except $$x = \frac{1}{3}$$

Question 4

Find the domain of $$f(x) = \frac{2x^2$ - 5x + 3}{x^2$ + x - 6}.$$

Accepted Answer

All real numbers except $$x = 2$$ and $$x = -3$$

Question 5

Find the equation of the line passing through $$(-5, 4)$$ and $$(3, -2) in $$slope-intercept form.

Accepted Answer

$$y = -\frac{3}{4}x - \frac{11}{4}$$

Question 6

Find the equation of the line passing through $$(-1, 3)$$ that is parallel to the line $$2x + 6y = -6. $$Express your answer in slope-intercept form.

Accepted Answer

$$y = -\frac{1}{3}x + \frac{8}{3}$$

Question 7

Find the equation of the line passing through $$(-1, 3)$$ that is perpendicular to the line $$2x + 6y = -6. $$Express your answer in slope-intercept form.

Accepted Answer

$$y = 3x + 6$$

Question 8

A circle has center $$(-1, 2)$$ and passes through the point $$(4, 3). $$What is its equation?

Accepted Answer

$$(x + 1)^2$ + (y - 2)^2$ = 26$$

Question 9

Given the circle $$(x + 1)^2$ + (y - 4)^2$ = 2$$, what is its center and radius?

Accepted Answer

Center $$(-1, 4)$$, radius $$\sqrt{2}$$

Question 10

Given $$f(x) = 5x + 2$$ and $$g(x) = x^2$ - x$$, find $$f(g(x))$$ and its domain.

Accepted Answer

$$f(g(x)) = 5(x^2$ - x) + 2 = 5x^2$ - 5x + 2$$, 
Domain: All real numbers

Question 11

If $$f(x) = x^2$ - 1$$ and $$g(x) = x^2$ + 5$$, find $$(f \circ g)(x)$$ and its domain.

Accepted Answer

$$(f \circ g)(x) = (x^2$ + 5)^2$ - 1$$, 
Domain: All real numbers

Question 12

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

Accepted Answer

$$f(g(x - 2)) = ((x - 2) + 1)^2$ - 1 = (x - 1)^2$ - 1 = x^2$ - 2x$$

Question 13

If $$f(x) = x^2$ + 2$$ and $$g(x) = x^2$ - 4$$, find $$(f \circ g)(-2).$$

Accepted Answer

$$(f \circ g)(-2) = f(g(-2)) = f(0) = 0^2$ + 2 = 2$$

Question 14

Describe the sequence of transformations to graph $$f(x) = |2x + 3|$$ starting from $$y = |x|.$$

Accepted Answer

First, horizontally compress by factor $$\frac{1}{2}$$, then shift left by $$\frac{3}{2}$$ units.

Question 15

A chemical reaction's rate is modeled by $$f(x) = \frac{1}{x - 2}. $$What is the domain of $$f$$?

Accepted Answer

All real numbers except $$x = 2$$

Question 16

A medical dosage is modeled by $$f(x) = \sqrt{x - 5}. $$What is the domain of $$f$$?

Accepted Answer

$$x \geq 5$$

Question 17

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

Accepted Answer

$$f(g(x - 2)) = (x - 1)^2$ - 1 = x^2$ - 2x$$

Question 18

A population model is given by $$f(x) = 2|x| - 3. $$Which transformation is applied to $$y = |x| to $$obtain $$f(x)$$?

Accepted Answer

Vertical stretch by factor $$2$$, then shift down by $$3$$

College Algebra Exam 1 Review: Step-by-Step Guidance

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