🧮 algebra

1. **صورت مسئله را می‌نویسیم.** می‌خواهیم بدانیم خط افقی $y=k$ برای چند عدد صحیحِ $k$ نمودار تابع زیر را **در دقیقاً یک نقطه** قطع می‌کند:

1. مسئله را بیان می‌کنیم: تابع چندضابطه‌ای داده شده است: $$f(x) = \begin{cases} x^2 + a & x \geq a \\ 3x - a & x \leq a \end{cases}$$

1. **صورت مسئله:** تابع $f(x)$ همانی است، یعنی برای هر $x$ داریم $f(x)=x$.

1. مسئله را بیان می‌کنیم: تابع $$f(x)=\frac{(x-2)^2 - x(x+1)}{mx+3}$$ با دامنه $$\mathbb{R} \setminus \left\{-\frac{3}{m}\right\}$$ داده شده است و برد آن برابر با مجموعه $$\{a\}$$

1. The problem asks to identify which function corresponds to the given graph. 2. The graph is described as an increasing cube-root-shaped curve with an inflection point around $(2

1. **State the problem:** We are given two functions $f(x) = 2x^2 + 1$ and $g(x) = 2x - 6$. We need to determine which of the statements A through E about $f$ and $g$ are true. 2.

1. **State the problem:** Evaluate the expression $-11 + 5 - (-9) + 2$. 2. **Recall the rule for subtracting negative numbers:** Subtracting a negative number is the same as adding

1. Let's clarify the problem: You are asking why a certain value is not equal to -4. 2. To answer this, I need to know the exact expression or equation you are referring to.

1. **State the problem:** We are given two quadratic functions $a(x) = x^2 + 3x + 1$ and $b(x) = (x + 3)(x + 1)$ and asked to evaluate each at chosen $x$-values: $-2, -1, 0, 1, 2$.

1. **State the problem:** Solve the quadratic equation $$x^2 - 7x + 10 = 23$$. 2. **Rewrite the equation:** Move all terms to one side to set the equation equal to zero:

1. **State the problem:** Solve the quadratic equation $$x^2 - 7x + 10 = 28$$. 2. **Rewrite the equation:** Move all terms to one side to set the equation to zero:

1. **Problem Statement:** Solve the system of linear equations. Since you did not provide specific equations, I will explain the general method for solving linear systems. 2. **For

1. **State the problem:** We are given a set of data points representing the height $y$ of a rocket at different times $x$ in seconds. We need to write a quadratic regression equat

1. **State the problem:** We are given a quadratic function $f(x) = ax^2 + bx + c$ and some values of $f(x)$ at specific $x$ values. We need to find $f(1)$. 2. **Write down the giv

1. Énonçons le problème : Calculer la dérivée des fonctions g(t), h(x) et y données. 2. Rappel de la règle de dérivation d'un quotient :

1. The problem asks for the basic formula, but it is not specified which formula is needed. 2. In algebra, a common basic formula is the quadratic formula used to solve equations o

1. The problem is to solve the equation $2x + 3 = 11$ for $x$. 2. The formula used here is to isolate $x$ by performing inverse operations. We subtract 3 from both sides and then d

1. The problem asks to find the vertex, axis of symmetry, and y-intercept for given quadratic functions and verify your answers. 2. For #1, you wrote:

1. **Problem Statement:** Verify the horizontal asymptote, domain, continuity, holes, vertical asymptotes, and intercepts for the functions: - $f(x) = \frac{5x^2+125}{4x^3-8x^2-9x+

1. The problem is to determine if $a = 500$ given that $y = 500$ when $c = 0$. 2. Typically, in a linear equation of the form $y = ac + b$, when $c = 0$, $y$ equals the constant te

1. Let's clarify how to identify the answers $a$ and $b$ from the previous worksheet. 2. Typically, $a$ and $b$ represent variables or solutions to equations given in the problems.