Practice Test 36 - Algebra 2

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

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1. Let $z_1 = 6 - 2i$ and $z_2 = 1 + 5i$. What is $z_1 - z_2$ in standard form $a+bi$?
2. Find the inverse function $f^{-1}(x)$ for $f(x)=\dfrac{4}{x+1}$ with domain restriction $x \ne -1$.
3. Solve $6\cdot 2^{2t-1}=100$ for $t$. Express your answer as a logarithm (base 2) and approximate using technology.
4. The inverse of matrix $A = \begin{pmatrix} 2 & 1 & 0 \\ 1 & 2 & 1 \\ 0 & 1 & 2 \end{pmatrix}$ contains the entry $\frac{3}{4}$ in position $(1,1)$. If this matrix is used to solve the system $A\vec{x} = \begin{pmatrix} 8 \\ 6 \\ 4 \end{pmatrix}$, and the solution has integer coordinates, what is the sum of all coordinates in the solution vector?
5. Write the expression as a single logarithm (condense using log properties): $2\log_{10}(a)+\tfrac{1}{2}\log_{10}(b)-\log_{10}(c)$ (Assume $a>0$, $b>0$, and $c>0$.)
6. A system of linear equations is given by: $3x - 2y + z = 7$ $x + 4y - 3z = -2$ $-2x + y + 5z = 8$ When this system is written in the matrix form $A\vec{x} = \vec{b}$, what is the entry in the second row, third column of matrix $A$?
7. Finite vs. infinite: Consider the geometric series $S_n=9+9\left(\frac{1}{4}\right)+9\left(\frac{1}{4}\right)^2+\cdots+9\left(\frac{1}{4}\right)^{n-1}.$ Which statement is true?​
8. The temperature of a cooling object follows $T(t) = 75 + 125(0.8)^t$, where $T(t)$ is temperature in °F and $t$ is time in minutes. What is the practical meaning of the parameter $75$ in this model?
9. Demonstrate that $f(x)=7-2x$ and $g(x)=\dfrac{7-x}{2}$ are inverse functions by showing both $f(g(x))=x$ and $g(f(x))=x$.
10. Consider the function $s(x) = 4x^3 + 3$. To find the value $c$ such that $s(c) = 31$, which equation must be solved?
11. Let $f(x)=x^2-2$ and $g(x)=2x+1$. Use the intersection points of the graphs $y=f(x)$ and $y=g(x)$ to solve $f(x)=g(x)$. Which set of $x$-values are the solutions?
12. What are the period and amplitude of $t(x)=2\cos\left(\tfrac{1}{2}x\right)$?​
13. A student saves money by depositing $25 each week into a jar and does not remove any money. After the first week, the jar has $25. Let $S_n$ be the total saved after $n$ weeks (integer domain $n\ge 1$). Write both a recursive and an explicit formula for $S_n$.
14. Let $z=2-5i$. Its conjugate is $\overline{z}=2+5i$. How are the points representing $z$ and $\overline{z}$ related geometrically on the complex plane?
15. Verify or disprove: $f(x)=\dfrac{x}{4}-3$ and $g(x)=4x-12$ are inverses. Use composition to check whether $f(g(x))=x$ and $g(f(x))=x$.​
16. Solve $x^2-8x+16=0$ in the complex numbers. Recall: every quadratic $ax^2+bx+c=0$ has exactly 2 complex solutions counting multiplicity. When $b^2-4ac=0$, there is one repeated real solution with multiplicity 2. Which answer correctly lists the solution(s) and multiplicity?
17. Function $f$ is given algebraically by $f(x)=-2(x+1)^2+8$. Function $g$ is described verbally as: "$g$ is a parabola that opens downward and has a maximum value of $10$." Which function has the larger maximum value?
18. Solve by taking square roots: $(2x+1)^2 = 36$
19. Generate a Pythagorean triple using the identity $(x^2 + y^2)^2 = (x^2 - y^2)^2 + (2xy)^2$ with $x=4$ and $y=1$. Which triple $(a,b,c)$ is produced?
20. A function is defined by $f(x)=3x+2$. Consider equal intervals of length 1. Which table of values and differences correctly demonstrates that $f$ has constant differences (and therefore forms an arithmetic sequence when sampled at integers)?

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