Question 1

A company is designing a rectangular garden such that its area is $$48$$ square meters and its length is $$4$$ meters longer than its width. Which quadratic equation can be used to find the width $$w of $$the garden, and what is the width?

Accepted Answer

The equation is $$w^2 + 4w - 48 = 0$; the width is w = 6$ meters.

Question 2

Solve the equation $$\sqrt{x + 5} = x - 1$$ for $$x.$$

Accepted Answer

$$x = 4$$

Question 3

Which of the following relations is NOT a function?

Accepted Answer

The set $$\{(1,2), (2,3), (3,4), (1,5)\}$$

Question 4

Given the function $$f(x) = |2x - 6|$$, for which value of $$x$$ does $$f(x) = 0$$?

Accepted Answer

$$x = 3$$

Question 5

A bridge is designed in the shape of a parabola described by $$y = -0.5(x - 10)^2$ + 25. $$What are the coordinates of the vertex, and is it a maximum or minimum?

Accepted Answer

Vertex at $$(10, 25)$$; maximum point.

Question 6

Find the equation of the line perpendicular to $$y = 2x + 3$$ that passes through the point $$(4, 1).$$

Accepted Answer

Question 7

Given $$f(x) = x^2$ - 4$$ and $$g(x) = \sqrt{x}$$, find $$(f \circ g)(16).$$

Accepted Answer

$$(f \circ g)(16) = 0$$

Question 8

Which of the following is the inverse of the function $$f(x) = 3x + 7$$?

Accepted Answer

$$f^{-1}(x$$) = \frac{x - 7}{3}$$

Question 9

A population of bacteria triples every 4 hours. If the initial population is $$200$$, which equation models the population $$P$$ after $$t$$ hours?

Accepted Answer

$$P = 200 \cdot 3$$^{t/4}$$

Question 10

Solve for $$x$$: $$2^{x+1} = 16$.

Accepted Answer

$$x = 3$$

Question 11

Which of the following is equivalent to $$\log_2$ 32$$?

Accepted Answer

$$5$$

Question 12

Write $$\log_{10} 1000 as a $$sum of logarithms using properties of logarithms.

Accepted Answer

$$\log_{10} 10 + \log_{10} 10 + \log_{10} 10$$

Question 13

Find the domain of the rational function $$f(x) = \frac{x+2}{x^2$ - 9}.$$

Accepted Answer

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

Question 14

Which of the following equations represents a line parallel to $$y = -\frac{1}{3}x + 2$$ and passing through $$(0, -4)$$?

Accepted Answer

$$y = -\frac{1}{3}x - 4$$

Question 15

If $$f(x) = x^2$ - 6x + 8$$, what is the vertex of the parabola?

Accepted Answer

$$(3, -1)$$

Question 16

Solve the absolute value inequality $$|2x - 5| \u003c 7.$$

Accepted Answer

$$-1 \u003c x \u003c 6$$

Question 17

Which of the following statements is TRUE about the function $$f(x) = x^3$ - 2x^2$ + x$$?

Accepted Answer

The graph can have up to $$2$$ turning points.

Question 18

Given $$f(x) = 2x + 1$$ and $$g(x) = x^2$, find the domain of (f$$ \circ g)(x)$$.

Accepted Answer

All real numbers

Question 19

Solve the logarithmic equation $$\log_3$(x - 2) = 2.$$

Accepted Answer

$$x = 11$$

Question 20

A radioactive substance decays according to $$A(t) = 100e$$^{-0.05t}$$, where t$ is in years. How much of the substance remains after 20$ years?

Accepted Answer

$$A(20) = 100e$$^{-1}$$ \approx 36.79$$

MAC 1105H College Algebra Honors: Comprehensive Study Guide

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