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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