Practice Test 81 - Algebra 2

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

Given $f(x) = \frac{x - 2}{3x + 1}$ with domain $x \neq -\frac{1}{3}$ , and $g(x) = \frac{-x - 2}{3x - 1}$ with domain $x \neq \frac{1}{3}$ , a student computes $f(g(x))$ and $g(f(x))$ and finds both equal $x$ . However, the functions are still not inverses. What is the most likely explanation?
Simplify the expression (combine like terms). Because polynomials are closed under addition, subtraction, and multiplication, the simplified result is a polynomial: $(x+2)^2-(x-1)(x+3).$
The surface area formula for a cylinder is $S = 2\pi r^2 + 2\pi rh$ . A manufacturer wants to express the height in terms of the other variables to optimize production costs. Which rearrangement correctly isolates $h$ ?
Distinguish between terms and factors: In the expression $2x(x-3)^2 + 5$ , which list correctly gives the terms, and also gives the factors of the first term?
A medication dose starts at 240 mg. Each hour, 20% of the medication is eliminated (so 80% remains). Model the amount remaining with an exponential equation and solve for how many hours $t$ it takes until only 60 mg remains. Let $t$ = time in hours.
A model for the height of a ball (in feet) $t$ seconds after it is thrown is $h(t)=-16t^2+48t+5$ . The ball is in the air from the moment it is thrown until it hits the ground at $t=3$ seconds. What domain makes sense for $t$ in this context?
Multiple structural views are possible. Use structure to factor $x^4-1$ completely over the integers. Which is a complete factorization of $x^4-1$ ?
How many real solutions does this system have? (Graphically, this is the number of intersection points of the line and parabola.) $\begin{cases} y = x^2 + 4 \\ y = 2x + 1 \end{cases}$
Find the inverse function $f^{-1}(x)$ for $f(x)=\sqrt{x-2}+1$ , with domain $x\ge 2$ .
Order these functions by eventual growth rate (fastest to slowest) as $x\to\infty$ : - $f(x)=0.1\cdot 4^x$ (exponential) - $g(x)=x^7$ (polynomial) - $h(x)=500x^2$ (quadratic) - $p(x)=12x$ (linear)
Solve the equation by completing the square: $x^2+4x-5=0.$
A company's profit $P$ (in thousands of dollars) depends on the number of units sold $x$ (in hundreds) according to a quadratic relationship. Based on the coordinate plane shown, which equation best represents this relationship?
Solve the quadratic equation by completing the square: $x^2+8x+7=0.$ Write your work in the form $(x-p)^2=q$ and then give the solutions for $x$ .
Let $f(x)=\sqrt{x}$ . Which function represents a horizontal shift of the graph of $f$ right by 3 units?
Derive the equation of a circle with center $(4, -1)$ and radius $3$ .
A streaming service charges a base monthly fee plus an additional amount per premium channel. The cost is $C = 6p + 12$ , where $C$ is in dollars and $p$ is the number of premium channels. In the function $C = 6p + 12$ , what does the y-intercept (the parameter $12$ ) represent?
Find and verify the inverse of $f(x)=\dfrac{2x+3}{5}$ (the inverse should satisfy $f(f^{-1}(x))=x$ ).
Multiply and simplify using $i^2 = -1$ : $(2 - 3i)(4 + i)$ . Write your answer in $a + bi$ form.
What is the exponential form of $\log_2(x)=7$ ?
Use synthetic division (more efficient than long division for $(x-c)$ ) to divide $P(x)=x^3-2x^2-5x+6$ by $(x-3).$ Use $c=3$ and coefficients $1,-2,-5,6$ . Which result is correct? (Remainder Theorem check: remainder $=P(3)$ .)

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