Practice Test 65 - Algebra 2

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

Over the complex numbers, the difference of squares identity $a^2-b^2=(a+b)(a-b)$ still holds. Factor the polynomial $x^2+9$ completely over $\mathbb{C}$ .
Use the identity $(x^2+y^2)^2=(x^2-y^2)^2+(2xy)^2$ to generate a Pythagorean triple when $x=3$ and $y=2$ . Which triple $(a,b,c)$ results, where $a^2+b^2=c^2$ ?
The function $f(x)=x^2-4x+3$ has domain all real numbers and fails the horizontal line test. What domain restriction makes $f$ invertible?
Verify that the functions $f(x)=4x-9$ and $g(x)=\dfrac{x+9}{4}$ are inverse functions by composition. Compute and simplify both $f(g(x))$ and $g(f(x))$ , and determine whether both equal $x$ .
A tank is being filled by two hoses. Hose A can fill the tank in 6 hours and Hose B can fill the same tank in 9 hours. Write and solve an equation to find how long it takes to fill the tank if both hoses run together. Let $t$ = the number of hours to fill the tank together.
A delivery company is packing boxes with two item types. Let $x$ = number of Type A items and $y$ = number of Type B items. A box can hold at most 30 items total: $x+y\le 30$ . Type A items weigh 2 lb each and Type B items weigh 1 lb each, and the box can weigh at most 48 lb: $2x+y\le 48$ . Also, at least 12 Type B items must be included: $y\ge 12$ . Which point is viable?
An investment account follows the model $A(t)=2500(1.06)^t$ , where $t$ is in years. Identify the constant percent rate and classify the change.
Determine whether the data are best modeled by a linear or exponential function, and write a function that models the data. Table: $x: 0,\ 1,\ 2,\ 3$ and $f(x): 5,\ 10,\ 20,\ 40$
Use the logarithm properties $\log_b(xy)=\log_b(x)+\log_b(y),\quad \log_b\left(\frac{x}{y}\right)=\log_b(x)-\log_b(y),\quad \log_b(x^p)=p\log_b(x)$ to expand the expression $\log_3\left(\frac{x^4y}{z^2}\right)$ as a sum and/or difference of logarithms (no logarithms of products, quotients, or powers should remain).
A manufacturing process uses three machines to produce items with different resource requirements. The system $\begin{pmatrix} 2 & 3 & 1 \\ 1 & 1 & 2 \\ 3 & 2 & 1 \end{pmatrix}\begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} 23 \\ 14 \\ 20 \end{pmatrix}$ represents the resource allocation. If the inverse matrix method gives $x = 3$ , $y = 5$ , $z = 2$ , but resource availability changes so that the first constraint increases to $28$ while others remain the same, how does the solution change?
A communication tower is located at $(3,4)$ . Its signal reaches all points within a radius of $10$ miles. What is the equation of the circle modeling the boundary of the coverage area?
A quantity decays by 10% per year, so its value after $t$ years is multiplied by $(0.90)^t$ . Which expression correctly rewrites $(0.90)^t$ to show an equivalent daily factor raised to the number of days? (Assume 365 days in a year.)
A polynomial $F(x) = 20x^5 + ax^4 + bx^3 + cx^2 + dx + 63$ has exactly three rational zeros. Using the Rational Zero Theorem, what is the maximum number of negative rational zeros that $F(x)$ could have?
The function $f(x)=-x^2+10$ is not invertible on all real numbers because it fails the horizontal line test. Which restriction makes $f$ invertible?
For the polynomial $p(x)=-x^3+6x^2-9x,$ factor to find the zeros and state the end behavior (enough to sketch).
A bacterial culture is modeled by $P(t) = 300\cdot 1.5^t$ , where $P$ is the number of bacteria and $t$ is time in hours. In this model, what does the parameter $300$ represent?
A system of linear equations has exactly one solution at $(-2, 5)$ . If the first equation is $y = 3x + 11$ , which of the following could be the second equation?
Use conjugates to rationalize the denominator and write in $a+bi$ form: $\frac{2-i}{3+2i}$
Let the parent function be $f(x)=x^2$ , whose graph is the standard parabola with vertex at $(0,0)$ . What transformation changes $f(x)$ to $g(x)=2f(x-3)+1$ ?
A medication level in the bloodstream is modeled by $M(t) = 150(0.75)^t$ , where $t$ is measured in hours after the dose and $M$ is measured in milligrams. In $M(t) = 150(0.75)^t$ , what does it mean that the base is $0.75$ ?

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