Practice Test 29 - Algebra 2

20 questions. Answer them in any order, then submit to see your score and every explanation.

Demonstrate by composition that $f(x)=\dfrac{1}{x-4}$ and $g(x)=4+\dfrac{1}{x}$ are inverse functions (where defined). Compute $f(g(x))$ and $g(f(x))$ and simplify to see whether each equals $x$ .
Identify the perfect-square structure and rewrite the expression. Which rewrite is correct for $x^2-10x+25$ ?
On the complex plane, the distance from the origin to the point representing $z=a+bi$ equals $|z|=\sqrt{a^2+b^2}$ . What is the distance from the origin to the point representing $z=-5-12i$ ?
Which expression represents the most complete simplification of $\log_3\left(\frac{9x^4}{\sqrt{27y^2}}\right)$ using logarithm properties?
The value of a laptop over time is modeled by $V(t)=900(0.80)^t$ , where $t$ is the number of years since purchase and $V$ is in dollars. What does the parameter $900$ represent in context?
Solve the system using the matrix inverse method. First write $AX=B$ , then compute $X=A^{-1}B$ . $\begin{cases} 2x+y=5 \\ 3x+4y=7 \end{cases}$ What is $(x,y)$ ?
A rectangular garden has a perimeter of 120 feet. If the length is represented by $l$ and the width by $w$ , and the area must be at least 800 square feet, which system of equations and inequalities correctly models this situation?
An investment account starts with $1000 and earns $4\%$ interest compounded annually. Create an exponential model for the balance $A(t)$ after $t$ years, then solve for when the account balance doubles. Round to the nearest tenth of a year and interpret your result.
Let the parent function be $f(x)=|x|$ . What is the effect on the graph of $g(x)=f\!\left(\tfrac{1}{2}x\right)$ ?
Find the center and radius of the circle given by $x^2+y^2+6x-4y+9=0$ .
Consider the system of equations: $2x + 3y = 8$ $5x - y = 1$ If this system is represented as $A\vec{x} = \vec{b}$ where $\vec{x} = \begin{pmatrix} x \\ y \end{pmatrix}$ , which of the following correctly represents the matrix equation?
On the complex plane, the horizontal axis is the real axis and the vertical axis is the imaginary axis. The complex number $z=3+2i$ is represented by the point $(a,b)$ where $a$ is the real part and $b$ is the imaginary part. Which point represents $z$ , and what is its modulus $|z|=\sqrt{a^2+b^2}$ ?
Solve and evaluate: $12\cdot 10^{t/3}=90$ . Give $t$ in exact logarithmic form and as a decimal (use a calculator).
What is the 4th term (counting the first term as term 1) in the expansion of $(x+y)^6$ using Pascal's Triangle coefficients?
The matrix equation $\begin{bmatrix}2&-1\\3&4\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}5\\7\end{bmatrix}$ represents what system of equations?
A community garden is deciding how many acres of tomatoes and peppers to plant. Let $x$ be acres of tomatoes and $y$ be acres of peppers. Constraints: - Total land available: at most 12 acres: $x+y\le 12$ . - Water limit: tomatoes need 3 units/acre and peppers need 2 units/acre; at most 30 units: $3x+2y\le 30$ . - The garden must plant at least 2 acres of peppers: $y\ge 2$ . - Non-negativity: $x\ge 0$ , $y\ge 0$ . Which point is viable?
The value of a car depreciates according to the formula $V(t) = 25000e^{-0.15t}$ , where $t$ is time in years. In how many years will the car be worth exactly half its original value?
Solve the equation and write the solutions in $a\pm bi$ form: $x^2 + 4x + 8 = 0$ .
Compound interest is compounded annually. You deposit $2500 into an account earning $5\%$ interest per year. Create an exponential model for the balance $A(t)$ after $t$ years, then solve for how long it will take the balance to reach $4000. Round your time to the nearest tenth of a year and interpret your result in context.
What are the asymptotes of $f(x)=\frac{x^2}{x^2-9}?$

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