Question 1

Given the augmented matrix in row-echelon form:

$$
\begin{bmatrix}
1 & -1 & 2 & | & -1 \
0 & 1 & -5 & | & 3 \
0 & 0 & 1 & | & \frac{13}{2}
\end{bmatrix}
$$

What is the solution to the associated system of equations for variables $$x$$, $$y$$, and $$z$$?

Accepted Answer

$$(x, y, z) = (3, -3, -4)$$

Question 2

Solve the following system using the elimination method:

$$
\begin{cases}
6x - 5y = 42 \
5x + 30y = -170
\end{cases}
$$

What is the solution?

Accepted Answer

$$(x, y) = (3, -6)$$

Question 3

Which graph best represents the solution set to the system of inequalities $$x + y  6$$?

Accepted Answer

There is no solution region.

Question 4

Gaby's piggy bank contains nickels and quarters worth $4.75. If she has 31 coins in all, how many nickels does she have?

Accepted Answer

12

Question 5

Classify the function $$h(t) = \frac{4 - 8t}{6t^2$}$$ and state its domain.

Accepted Answer

Rational function; $$(-\infty, 0) \cup (0, \infty)$$

Question 6

Given the matrix:

$$
\begin{bmatrix}
0 & 6 & -6 \
8 & 5 & 8 \
8 & 9 & 2
\end{bmatrix}
$$

What is the size and type of this matrix?

Accepted Answer

3 × 3, square matrix

Question 7

For the function $$f(x) = 2x^2$ - 10x - 72$$, what are the $$x-$$intercepts and $$y-$$intercept?

Accepted Answer

$$x-$$intercepts: $$9$$ and $$-4.0$$; $$y-$$intercept: $$-72$$

Question 8

Determine if the quadratic function $$f(x) = \frac{1}{2}x^2$ + 2x - 1$$ has a maximum or minimum value and find that value.

Accepted Answer

Minimum at $$(-2, -3)$$

Question 9

The cost $$C(x) of $$operating a video store is given by $$C(x) = 2x^2$ - 32x + 790$$, where $$x is $$the number of videos rented daily. What is the lowest cost to the nearest dollar?

Accepted Answer

$662

Question 10

Find the inverse of the one-to-one function $$f(x) = \frac{5}{3x + 2}.$$

Accepted Answer

$$f^{-1}(x$$) = \frac{5 - 2x}{3x}$$

Question 11

For the polynomial $$f(x) = -6(x-5)^2$(x^2-25$$)$$, what are the real zeros and their multiplicities? Does the graph touch or cross the x$-axis at each zero?

Accepted Answer

Zeros: $$-5 ($$multiplicity 1, touches), $$5 ($$multiplicity 3, crosses)

Question 12

Expand the logarithmic expression using properties of logarithms:

$$
\log_4$ \left( \frac{16x^4$}{6 \cdot 4y^7$} \right)
$$

Accepted Answer

$$2 + 2\log_4$ x - (\log_4$ 6 + 1 + 7\log_4$ y)$$

Question 13

Solve for $$x$$:

$$
\log(4x) = \log 5 + \log(x-2)
$$

Accepted Answer

$$x = 10$$

Question 14

Approximate the solution to the exponential equation $$e^{x-2}$$ \cdot $$e^{2x+8} = 29$ to four decimal places.

Accepted Answer

$$x \approx -0.8776$$

Question 15

Given the rational function $$f(x) = \frac{x}{x^2$ + 2x}$$, identify the coordinates of all removable discontinuities and the equations of all asymptotes.

Accepted Answer

Removable discontinuity at $$(0, \frac{1}{2})$$; vertical asymptote at $$x = -2$$; horizontal asymptote at $$y = 0$$

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