Practice Test 94 - Algebra 2

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

Verify by composition whether $f(x)=\dfrac{x-3}{x+2}$ and $g(x)=\dfrac{2x+3}{1-x}$ are inverses. Compute $f(g(x))$ and $g(f(x))$ and check if both equal $x$ (where defined).
If $\log_3(x - 1) + \log_3(x + 2) = 2$ , which equation results from applying the product property of logarithms?
The function $f(x)=x^4$ is not invertible on all real numbers because it fails the horizontal line test (for example, $f(1)=f(-1)$ ). Which domain restriction makes $f$ invertible?
A quick way to divide a polynomial by a linear divisor $(x-c)$ is synthetic division, which is usually faster than long division. Use synthetic division to divide $P(x)=x^3+4x^2-3x+7$ by $(x-2).$ (Use $c=2$ on the left and the coefficients $1,4,-3,7$ on the top. The bottom row gives the quotient coefficients, and the last value is the remainder, which also equals $P(2)$ by the Remainder Theorem.) What are the quotient and remainder? Write your result in the form $P(x)=(x-2)Q(x)+R$ .
A city's population model is $P(t) = 125000 + 2800t$ , where $t$ represents years since 2020. Urban planners want to determine when the population will exceed 140000. However, they realize the growth rate may be 15% lower than modeled. What is the difference in predicted timing between the original and adjusted models?
Simplify using $i^2=-1$ : $i^3$ .
A phone plan charges a $15 monthly fee plus $0.10 per text message. Let $C$ be the total cost (in dollars) for $x$ text messages. Is the rate of change of $C$ with respect to $x$ constant? If so, what is it and what type of function models the situation?
A store sells two sizes of juice: small ( $s$ ) and large ( $\ell$ ). The totals satisfy $\begin{cases} 2s+\ell=11\\ -s+3\ell=4 \end{cases}$ Write the system as $AX=B$ using variable order $s, \ell$ .
Over the complex numbers, the difference of squares identity $a^2-b^2=(a+b)(a-b)$ still applies. Which option correctly factors $x^2+9$ over $\mathbb{C}$ ?
A rectangle has length $(x+4)$ meters and width $(x-1)$ meters. 1) Write an equation for the area $A$ (square meters) in terms of $x$ . 2) Then choose appropriate axes labels and a reasonable scale to graph $A$ versus $x$ for $1\le x\le 8$ . Which choice is correct?
Verify that the functions $f(x)=3x-5$ and $g(x)=\dfrac{x+5}{3}$ are inverse functions by composition. Compute both $f(g(x))$ and $g(f(x))$ and determine whether each equals $x$ .
A radio tower is located at $(3,4)$ on a coordinate grid. Its signal reaches all points within 10 units of the tower. Write the equation of the circle representing all points exactly 10 units from the tower.
Use synthetic division to divide $P(x)=2x^3-7x^2+4x-3$ by $(x-3).$ Set up with $c=3$ and coefficients $2,-7,4,-3$ . The bottom row gives the quotient coefficients and the final number is the remainder (also $P(3)$ by the Remainder Theorem). Which result is correct?
Graph the polynomial function $p(x)=(x+2)^2(x-1)(x-4)$ showing all zeros (with whether the graph crosses or touches at each zero) and the correct end behavior. A rough sketch is sufficient.
A town's population is modeled by $P(t)=18{,}000(1.015)^t$ , where $t$ is time in years. What does it mean that the base is 1.015 in this model?
Let the parent function be $f(x)=\sqrt{x}$ . What is the effect on the graph of replacing $f(x)$ with $f\left(\frac{x}{2}\right)$ ?
The relativistic energy formula is $E = \frac{mc^2}{\sqrt{1 - \frac{v^2}{c^2}}}$ , where $E$ is energy, $m$ is rest mass, $c$ is speed of light, and $v$ is velocity. A physics student needs to solve for velocity $v$ . Which expression is correct?
A streaming service charges a one-time setup fee of $25 plus $8 per month. Create an inequality for a budget of at most $105 and find the maximum number of months you can pay for. Let $m$ = the number of months.
Use the inverse property $b^{\log_b(x)}=x$ (for $x>0$ ) to simplify $2^{\log_2(17)}$ .
A student is mixing Solution A (x liters) and Solution B (y liters) to make a lab mixture. - Total volume must be between 6 and 10 liters, inclusive. - Solution A must be at least 2 liters. - They cannot use negative amounts. Which system represents these constraints?

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