🧮 algebra

1. The problem defines a piecewise function: $$F(x) = \begin{cases} x+1, & |x| \leq 2 \\ 5 - x, & |x| > 2 \end{cases}$$

1. **State the problem:** We are given a piecewise function: $$F(x) = \begin{cases} x+1 & \text{if } |x| \leq 2 \\ 5 - x & \text{if } |x| > 2 \end{cases}$$

1. Stating the problem: Simplify the function $$F(x)=\frac{\sqrt{x^2+3-2}}{x-1}, \quad x\neq1$$. 2. Simplify inside the square root: $$x^2 + 3 - 2 = x^2 + 1$$.

1. The problem is to simplify the function $$F(x) = \frac{\sqrt{x^2 + 3 - 2}}{x - 1}, \quad x \neq 1.$$\n\n2. First, simplify the expression inside the square root: $$x^2 + 3 - 2 =

1. The problem is to analyze the function $$F(x) = \begin{cases} \frac{\sqrt{x^2 + 3 - 2}}{x - 1}, & x \neq 1 \\ 2, & x = 1 \end{cases}$$ 2. Simplify the expression inside the squa

1. We are given the function $$F(x) = \frac{\sqrt{x^2+3}}{x-1}$$ with the domain restriction $$x \neq 1$$ because the denominator cannot be zero. 2. To understand the behavior of t

1. The problem is to simplify the expression $$\frac{u_3 - u_2}{u_3 + u_2}$$. 2. This expression is a fraction where the numerator is the difference of two variables $u_3$ and $u_2

1. The problem asks to express the sum \(\frac{y+7}{1} + \frac{y+5}{1}\) as a single fraction in simplest form. 2. Since both fractions have denominator 1, we can add the numerator

1. The problem is to express the equation $$(2x + 1)(x - 3) = \frac{\square}{2x + 1} - \frac{\square}{x - 3}$$ by finding the numerators in the fractions on the right side. 2. Star

1. **State the problem:** Simplify the expression $$\frac{x-4}{7} \cdot \frac{x}{21}$$. 2. **Multiply the numerators and denominators:**

1. We are asked to express the sum $5 + \frac{1}{y}$ as a single fraction. 2. To combine these terms, we need a common denominator. The denominator here is $y$.

1. **Calcule a característica da matriz A.** A matriz A é dada por:

1. **State the problem:** Solve the equation $$4^x \cdot 16^{x+1} = \left( \frac{1}{2^{-5}} \right)^2$$ for $x$. 2. **Rewrite bases as powers of 2:**

1. **Problem 9:** Simplify $$\frac{(10^3 \cdot 20^2)^3}{5^5 \cdot 2} \cdot \frac{250^4 \cdot 8^3}{75^2}$$ 2. First, simplify inside the parentheses:

1. The problem asks to find the domain of two functions, labeled as question a and question b. 2. To find the domain of a function, identify all values of $x$ for which the functio

1. The problem asks for the domain of variables $a$ and $b$. 2. The domain of a variable is the set of all possible values it can take.

1. The problem is to simplify the expression $$\frac{1}{\sqrt{3} - 1}$$. 2. To simplify, multiply numerator and denominator by the conjugate of the denominator $$\sqrt{3} + 1$$ to

1. Problem a: Graph the function $f(x) = 5x^4 - 13$ on the window $[-2,2] \times [-16,16]$ and state its domain and range. 2. The domain of $f(x)$ is all $x$ values in the interval

1. The problem asks to fill in the blanks about functions and answer questions about independent and dependent variables. 2. A function is a rule that assigns to each value of the

1. **State the problem:** Simplify the expression $4y^3 + 16y$. 2. **Factor out the greatest common factor (GCF):** Both terms have a factor of $4y$.

1. **State the problem:** Simplify the expression $$\frac{18^3 \cdot 12^2 \cdot 128}{6^6 \cdot 24^{-5}}$$. 2. **Rewrite bases in prime factorization:**