🧮 algebra

1. **State the problem:** We are given three functions: $f(x) = 2x + 8$, $g(x) = x^2 + 2x - 8$, and $h(x) = 3x - 6$. We need to find the value of $f(0.5)$. 2. **Recall the formula:

1. The problem asks to find the domain of the function $$f(x) = \sqrt{-4x - 7}$$. 2. The domain of a square root function requires the expression inside the root to be non-negative

1. **State the problem:** We want to rewrite the expression $$(8.9 \times 10^8) - (4.4 \times 10^7)$$ so that both terms have the same power of 10, specifically $10^8$. 2. **Recall

1. **State the problem:** Given two functions $f(x) = 4 - x^2$ and $g(x) = 2 - x$, find the product function $(fg)(x)$, which means $f(x) \cdot g(x)$. 2. **Formula and explanation:

1. **State the problem:** Find the product of the functions $f(x) = 4 - x^2$ and $g(x) = 2 - x$, i.e., find $(fg)(x) = f(x) \cdot g(x)$.

1. **State the problem:** We have a manufacturing company with cost model $$C(x) = 0.35x^3 + 0.02x^2 + 4x + 240$$ and sales price model $$S(x) = 60 - 0.02x$$, where $x$ is the numb

1. **State the problem:** We are given two functions $f(x) = x^2 + 9$ and $g(x) = x + 3$. We need to determine which operation among addition, subtraction, multiplication, and divi

1. **State the problem:** We are given two functions: $f(x) = 2x + 3$ and $g(x) = 3[f(x)]^2 - 2$. We need to find the explicit form of $g(x)$ in terms of $x$. 2. **Recall the formu

1. **State the problem:** Given two functions $p(x) = ab^x$ and $r(x) = ca^x$, find the expression for $p(x) \cdot r(x)$. 2. **Recall the formula for multiplying exponential functi

1. **State the problem:** We are given two functions $S(s) = s^2 + s + 1$ and $T(s) = s - 5$. We need to find the sum $S(s) + T(s)$. 2. **Formula used:** To add two functions, we a

1. The problem asks to find the composition of functions $(fg)(x)$ where $f(x) = 1 - x^2$ and $g(x) = 1 - x$. 2. The composition $(fg)(x)$ means $f(g(x))$, which is the function $f

1. **Problem:** Given $f(x) = 2x^2 + 1$, find $g(x)$ where $g(x) = 2[f(x)]^2 - 1$. 2. **Formula:** To find $g(x)$, substitute $f(x)$ into the expression for $g(x)$:

1. **State the problem:** Solve the system of linear equations for $x$ and $y$: $$\begin{cases} 2x + y = 8 \\ 2x - y = 12 \end{cases}$$

1. The problem asks to find which matrices among A, B, C, and D are equivalent. 2. Two matrices are equivalent if they have the same dimensions and the same corresponding elements.

1. The problem is to combine the terms $-288x$ and $-288x$. 2. When combining like terms, you add or subtract their coefficients while keeping the variable part the same.

1. **State the problem:** Simplify the expression $4x \times -72$. 2. **Formula and rules:** Multiplying a variable term by a constant involves multiplying the constants and keepin

1. **State the problem:** We are given expressions for the sides of a quadrilateral and need to find the values of $a$ and $b$. 2. **Analyze the quadrilateral:** The quadrilateral

1. **State the problem:** Max wants to find out how many bottles of soda he must sell so that his total sales equal his total expenses. 2. **Define variables and costs:** Let $x$ b

1. **State the problem:** Two dogs start running toward a ball from different distances and speeds. We want to find the time when both dogs are the same distance from the ball.

1. **Problem Statement:** Consider the function $$f(x) = \frac{1}{x^2 + x - 6}$$.

1. **State the problem:** Find a polynomial function in factored form with zeros 2 (of multiplicity 2) and 3, passing through the point (1, 6). 2. **Write the general form:** Since