Q: An initial deposit of $10,000$ grows at an annual rate of $8\%$ for $7$ years. If the interest is compounded continuously, what is the amount in the account after $7$ years? (Use $A = Pe^{rt}$)

$A = 10,000 \times e^{0.08 \times 7} \approx 17,013$

Q: Find the inverse of the matrix $\begin{pmatrix} 1 & 2 & 3 \\ 0 & 1 & 2 \\ 0 & 0 & 1 \end{pmatrix}$.

$\begin{pmatrix} 1 & -2 & 1 \\ 0 & 1 & -2 \\ 0 & 0 & 1 \end{pmatrix}$

Question 1

Which of the following is the correct set of vertical asymptotes for the function $f(x) = \dfrac{2x-6}{x^2-x-12}$?

Accepted Answer

$x = 3,\ x = -4$

Question 2

Given $g(x) = \dfrac{3x^2+1}{3x^2-4x-4}$, what is the horizontal asymptote?

Accepted Answer

$y = 1$

Question 3

Which transformation is applied to the parent function $f(x) = 3^x$ to obtain $f(x) = -3^{x+1} + 2$?

Accepted Answer

Reflect over the $x$-axis, shift left 1 unit, shift up 2 units

Question 4

An initial deposit of $10,000$ grows at an annual rate of $8\%$ for $7$ years. If the interest is compounded continuously, what is the amount in the account after $7$ years? (Use $A = Pe^{rt}$)

Accepted Answer

$A = 10,000 	imes e^{0.08 	imes 7} \approx 17,013$

Question 5

Solve for $x$: $\log_2 8 = x$

Accepted Answer

$x = 3$

Question 6

Evaluate $\log_{10} 0.0001$ without a calculator.

Accepted Answer

$-4$

Question 7

Rewrite $\log \left( \sqrt{\dfrac{x^2 y}{z}} \right)$ in terms of $\log x$, $\log y$, and $\log z$.

Accepted Answer

$\log x + \dfrac{1}{2} \log y - \dfrac{1}{2} \log z$

Question 8

Solve for $x$: $3^{x+1} = 9^{2x}$

Accepted Answer

$x = -1$

Question 9

How long will it take for an account to double in value if the interest is compounded continuously at $8\%$ per year?

Accepted Answer

$t = \dfrac{\ln 2}{0.08} \approx 8.66$ years

Question 10

Solve the system of equations:

$\begin{cases} x + y + z = 3 \ x + 2y + z = 4 \ 2x + y + 2z = 5 \end{cases}$

Accepted Answer

$x = 1,\ y = 1,\ z = 1$

Question 11

Find the inverse of the matrix $\begin{pmatrix} 1 & 2 & 3 \ 0 & 1 & 2 \ 0 & 0 & 1 \end{pmatrix}$.

Accepted Answer

$\begin{pmatrix} 1 & -2 & 1 \ 0 & 1 & -2 \ 0 & 0 & 1 \end{pmatrix}$

Question 12

Find the determinant of the matrix $\begin{pmatrix} 0 & 2 & 0 & -2 \ 1 & 0 & 1 & 1 \ 0 & -1 & 0 & 1 \ 0 & 0 & 1 & 2 \end{pmatrix}$.

Accepted Answer

$-4$

Question 13

If $\log_5 x = 2$, what is the value of $x$?

Accepted Answer

$25$

Question 14

Solve for $x$: $\ln 10 = \ln(x-10) + \ln x$

Accepted Answer

$x^2 - 10x - 10 = 0$

Question 15

Which of the following is the correct solution for $5^{4x+6} = \dfrac{1}{125}$?

Accepted Answer

$x = -2$

Question 16

Given the system $\begin{cases} 3x + y = 7 \ x - y = 0 \ y - 2x = -8 \end{cases}$, what is the solution for $x$?

Accepted Answer

$x = 3$

Question 17

If $f(x) = \dfrac{3x^4 - 4x + 5}{2x}$, what is the end behavior of $f(x)$ as $x 	o \infty$?

Accepted Answer

$f(x) 	o \infty$

Question 18

Evaluate $\log_{e} e^{100}$.

Accepted Answer

$100$

College Algebra Practice Test 3: Rational Functions, Exponentials, Logarithms, and Systems of Equations

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