Practice Test 43 - Algebra 2

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

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1. Verify that the functions $f(x)=4x-7$ and $g(x)=\dfrac{x+7}{4}$ are inverses by composition. Compute both $f(g(x))$ and $g(f(x))$ and determine whether each equals $x$.
2. A savings account starts with $200$ dollars and increases by $10\%$ each month. Let $t$ be the number of months since the start. Write an exponential function $S(t)$ that models the balance.
3. A company models the average cost per item (in dollars) when producing $n$ items by $A(n)=\dfrac{500}{n}+12$. In context, $n$ is the number of items produced in a batch, and the factory can produce at most 200 items per batch. What is an appropriate realistic domain for $n$?
4. If the remainder when $p(x)=x^3-2x^2+kx-4$ is divided by $(x-2)$ is $6$, what is the value of $k$? (Use remainder $=p(2)$.)
5. Check if $f(x)=\dfrac{x+3}{2}$ and $g(x)=2x+3$ are inverse functions by computing both $f(g(x))$ and $g(f(x))$ and comparing each to $x$.
6. Use technology to solve the system by the inverse-matrix method. Write $AX=B$ and compute $X=A^{-1}B$. $\begin{cases} 2x+y+z=4 \\ x+3y+2z=5 \\ 3x+y+4z=6 \end{cases}$ What is $X=\begin{bmatrix} x \\ y \\ z \end{bmatrix}$?
7. Rewrite $\frac{3x^2-2x+4}{x-1}$ in the form $q(x)+\frac{r(x)}{x-1}$, where $\deg(r)<\deg(x-1)$. (You may use inspection or long division.)
8. A radioactive sample has a half-life of 12 years and initially has a mass of 160 grams. Create an exponential model for the remaining mass $N(t)$ after $t$ years, then solve for when 20 grams remain. Round your answer to the nearest tenth of a year and interpret your result.
9. A function is sampled at integer inputs, producing the sequence $7,\ 12,\ 17,\ 22$. Over equal input intervals of $1$, which statement correctly identifies the growth pattern and the most likely function type?
10. Write an expression for $f^{-1}(x)$ given $f(x)=x^3+5$.​
11. Use structure to factor the expression. Recognize it as a difference of cubes: $x^3 - 64$
12. Consider the exponential function $p(x)=2^x$ and the polynomial function $h(x)=x^4$. Using the table, at approximately what $x$-value does $2^x$ first exceed $x^4$?

    $x$	$x^4$	$2^x$
    14	38416	16384
    15	50625	32768
    16	65536	65536
    17	83521	131072
    18	104976	262144

13. A complex number $z$ has modulus 6 and argument $150°$. When $z$ is multiplied by the complex number $\frac{1}{2} - \frac{\sqrt{3}}{2}i$, what is the argument of the resulting product?
14. A medication dose starts at 180 mg and 20% is eliminated each hour. Model the amount remaining by an exponential equation and solve for how many hours until 60 mg remain. Let $t$ = time in hours.
15. For the arithmetic sequence $12,\ 7,\ 2,\ -3,\ \dots$ (viewed as a function with domain the positive integers), write both a recursive definition and an explicit formula for $a_n$.
16. Recognize and factor completely using polynomial identities: $x^4 - 16.$
17. Show that $f(x)=x^3+2$ and $g(x)=\sqrt[3]{x-2}$ are inverses by verifying that $f(g(x))=x$ and $g(f(x))=x$.​
18. If $r(x) = \sqrt[3]{2x + 7}$ and $r^{-1}(x)$ is the inverse function, which expression represents $r^{-1}(x)$?
19. Translate the recursive definition $a_1=48$ and $a_{n+1}=\tfrac{2}{3}a_n$ into an explicit formula for $a_n$ (for $n\ge 1$).
20. A pump can fill a tank in 6 hours, and a second pump can fill the same tank in 8 hours. Write and solve an equation to find how long it takes to fill the tank if both pumps run together. Let $t$ = time (in hours) to fill the tank together.

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