Q: Given the matrix $$\begin{pmatrix} 4 & -3 \\ 8 & 9 \\ 0 & -1 \\ -4 & 5 \end{pmatrix}$$ What is its size and how would you classify it?

Size: $$4 \times 2$$, None of these (not square, row, or column matrix)

Q: Let $$A = \begin{pmatrix} -7 & 9 \\ 0 & 3 \end{pmatrix}$$ and $$B = \begin{pmatrix} 6 & 0 \\ 0 & -8 \end{pmatrix}. $$Find $$A - \frac{1}{2}B.$$

$$\begin{pmatrix} 10 & -9 \\ 0 & -7 \end{pmatrix}$$

Question 1

Solve the system of equations using the elimination method: 
$$\begin{cases} -x + 2y = 0 \ x + 2y = 9 \end{cases}$$
What is the solution to the system?

Accepted Answer

$$\left( \dfrac{9}{2}, \dfrac{9}{4} \right)$$

Question 2

Benjamin & Associates built a total of 195 condominiums, each with either 3 or 4 bedrooms. If the total number of bedrooms is 693, how many three-bedroom units and four-bedroom units are there?

Accepted Answer

87 three-bedroom units and 108 four-bedroom units

Question 3

Solve the system using substitution or elimination: 
$$\begin{cases} x - 2y = 15 \ 3x - 6y = 15 \end{cases}$$
What is the solution?

Accepted Answer

There are an infinite number of solutions of the form $$(x, \frac{x-15}{2})$$

Question 4

Given the matrix 
$$\begin{pmatrix} 4 & -3 \ 8 & 9 \ 0 & -1 \ -4 & 5 \end{pmatrix}$$
What is its size and how would you classify it?

Accepted Answer

Size: $$4 	imes 2$$, None of these (not square, row, or column matrix)

Question 5

Let $$A = \begin{pmatrix} -7 & 9 \ 0 & 3 \end{pmatrix}$$ and $$B = \begin{pmatrix} 6 & 0 \ 0 & -8 \end{pmatrix}. $$Find $$A - \frac{1}{2}B.$$

Accepted Answer

$$\begin{pmatrix} 10 & -9 \ 0 & -7 \end{pmatrix}$$

Question 6

Given the augmented matrix in row-echelon form, use back substitution to solve the system:
$$\begin{pmatrix} 1 & -1 & 2 & | & -1 \ 0 & 1 & -\frac{13}{4} & | & 0 \ 0 & 0 & 1 & | & \frac{7}{2} \end{pmatrix}$$
What is the solution?

Accepted Answer

$$(1, -2, -3)$$

Question 7

Which graph best represents the solution set to the system of inequalities $$x + y \u003e -4$$ and $$x - y \u003c 3$$?

Accepted Answer

The first image (region in the upper right)

Question 8

Which graph best represents the solution set to the system $$x + 2y \leq 4$$ and $$y \geq \frac{x}{2} - 3$$?

Accepted Answer

The second image (bounded region between a parabola and a line)

Question 9

Classify the function $$f(x) = \frac{1}{x^2$ + 6x - 16}$$ and find its domain.

Accepted Answer

Rational function; $$(-\infty, -8) \cup (-8, 2) \cup (2, \infty)$$

Question 10

Given $$f(x) = 3x^2$ + 2$$ and $$g(x) = x^2$ - 16$$, find $$(f \div g)(x)$$ and its domain.

Accepted Answer

$$(f \div g)(x) = \frac{3x^2$ + 2}{x^2$ - 16}$$, domain: $$(-\infty, -4) \cup (-4, 4) \cup (4, \infty)$$

Question 11

Find the inverse of the one-to-one function $$f(x) = \sqrt{x + 3}.$$

Accepted Answer

$$f^{-1}(x$$) = $$x^2 - 3$

Question 12

Rewrite the quadratic function $$f(x) = -x^2$ - 4x + 12 in $$vertex form and find the $$x-$$intercepts.

Accepted Answer

Vertex form: $$f(x) = -(x + 2)^2$ + 16$$, $$x-$$intercepts: $$x = -6, 2$$

Question 13

You have 120 feet of fencing to enclose a rectangular plot that borders a river (no fence along the river). What dimensions maximize the area?

Accepted Answer

Length: 60 ft, Width: 30 ft

Question 14

Determine the real zeros and their multiplicities for $$f(x) = (x - 4)^2$ (x + 1)^3$. Does the graph touch or cross the x$-axis at each zero?

Accepted Answer

Zeros: $$x = -1 ($$multiplicity 3, crosses), $$x = 4 ($$multiplicity 2, touches)

Question 15

Expand the logarithmic expression as much as possible: $$\log_5$ \left( \frac{4x^3$ (2-x)}{4(x+2)^2$} \right)$$

Accepted Answer

$$\log_5$ 4 + 3\log_5$ x + \log_5$ (2-x) - \log_5$ 4 - 2\log_5$ (x+2)$$

Question 16

Condense the logarithmic expression: $$2\ln(x+3) - \ln x - \ln(x^2$ - 2)$$

Accepted Answer

$$\ln \left( \frac{(x+3)^2$}{x(x^2$-2)} \right)$$

Question 17

Solve the logarithmic equation: $$\log_6$(x+9) + \log_6$(x+44) = 2$$

Accepted Answer

$$x = -8$$

Question 18

Approximate the solution to the exponential equation $$8e^x \cdot e$$^{-x/5}$$ = 5 to $$four decimal places.

Accepted Answer

$$x \approx -0.5875$$

College Algebra Practice Final Exam Study Guide

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