🧮 algebra

1. **State the problem:** Find the horizontal asymptote(s) of the function $$f(x) = \frac{3x^4 + 4x + 4}{2x^4 + 3x - 3}$$. 2. **Recall the rule for horizontal asymptotes:**

1. **State the problem:** Find the horizontal asymptote(s) of the function $$f(x) = \frac{3x^4 + 2x + 2}{5x^4 + 2x - 4}$$. 2. **Recall the rule for horizontal asymptotes of rationa

1. **State the problem:** We are given a graph of a polynomial function with x-intercepts near $x \approx -9, -6, 4, 8$ and local extrema at $(-8,-5)$ (minimum), $(-2,9)$ (maximum)

1. The problem asks to find the value of $y$ for $y = f(4)$ using the graph of the function $f$. 2. From the description, the graph is a cubic curve crossing the x-axis near $-5$,

1. The problem asks to find the value of $x$ when $f(x) = 5$ using the graph of the function $f$. 2. From the graph, the horizontal line $y = 5$ intersects the curve at approximate

1. The problem asks to find the domain of the function $$g(x) = \sqrt{7 - x}$$. 2. The domain of a square root function requires the expression inside the root to be non-negative b

1. The problem is to graph the function $$y=2^x$$ over the interval $$[-3,3]$$ and identify the correct graph. 2. The function $$y=2^x$$ is an exponential growth function where the

1. The problem asks to describe the transformations to obtain the graph of $g(x) = 8^{x+3}$ from the graph of $f(x) = 8^x$. 2. The general form for horizontal shifts in exponential

1. **State the problem:** We need to find the amount $A$ using the formula for simple interest: $$A = P(1 + rt)$$ where $P$ is the principal, $r$ is the annual interest rate, and $

1. **State the problem:** Multiply the two given 2x2 matrices: $$\begin{bmatrix}-9 & -3 \\ -8 & -2\end{bmatrix} \times \begin{bmatrix}4 & 1 \\ 7 & -3\end{bmatrix}$$

1. **State the problem:** We are asked to perform the matrix multiplication of two matrices: $$\begin{bmatrix}-9 & -3 \\ -8 & -2\end{bmatrix} \times \begin{bmatrix}4 & 1 \\ 7 & -3\

1. **State the problem:** We need to graph the piecewise function $$f(x) = \begin{cases} 1 - 2x & \text{if } x < 1 \\ x - 3 & \text{if } x \geq 1 \end{cases}$$

1. **State the problem:** We need to graph the piecewise function $$f(x) = \begin{cases} 1 - 2x & \text{if } x < 1 \\ x - 3 & \text{if } x \geq 1 \end{cases}$$

1. **State the problem:** We need to graph the piecewise function: $$f(x) = \begin{cases} 3 - 2x & \text{if } x < 1 \\ x - 2 & \text{if } x \geq 1 \end{cases}$$

1. The problem asks how the graph of $g(x) = -|x - 5|$ is related to the graph of $f(x) = |x|$. 2. Recall the graph of $f(x) = |x|$ is a V-shaped graph with vertex at the origin $(

1. **State the problem:** Find the domain of the function $$f(x) = \frac{x - 9}{x + 1}$$. 2. **Recall the domain rule:** The domain of a function includes all real numbers except w

1. **State the problem:** Solve the quadratic equation $$x^2 + 8x = -4$$ for $x$. 2. **Rewrite the equation in standard form:** Move all terms to one side:

1. The problem asks us to complete the square for the polynomial $$x^2 - 6x + \_\_\_$$ to make it a perfect-square quadratic. 2. The formula to complete the square for an expressio

1. The problem asks whether the first step in completing the square is to double the middle term. 2. Completing the square is a method used to solve quadratic equations or rewrite

1. **State the problem:** Solve the quadratic equation $$w^2 - 2w - 13 = 0$$ by completing the square. 2. **Recall the formula and method:** To complete the square for an equation

1. The problem is to complete the square for the polynomial $x^2 + 12x + \_$ to make it a perfect-square quadratic. 2. The formula to complete the square for $x^2 + bx$ is to add $