🧮 algebra

1. Let's start by understanding what a vertex is in the context of a function, especially a quadratic function.\n\n2. The vertex of a parabola given by $y = ax^2 + bx + c$ is the p

1. The problem asks for the range of the function $$f(X) = \frac{2X + 1}{X - 1}$$ and also requests the vertex of this function. 2. First, let's understand the function type: $$f(X

1. **Problem Statement:** Find the range of the function $$f(x) = \frac{2x+1}{x-1}$$. 2. **Recall:** The range of a function is the set of all possible output values (values of $f(

1. The problem states that $f$ is an odd function, $f(3) = -2$, and the point $(-3, 5a - 13)$ lies on the curve of $f$. We need to find the value of $a$. 2. Recall the definition o

1. The problem is to solve the hardest question, but since no specific question was provided, I will demonstrate solving a challenging algebraic equation: Solve for $x$ in $$x^3 -

1. The problem is to solve algebraic equations. 2. Algebraic equations are mathematical statements that assert the equality of two expressions.

1. **Énoncé du problème :** Déterminer $f(0)$, $g(6)$, $f(-6)$, et $g \circ f(4)$ pour les fonctions $f(x) = x^2 - 2x + 2$ et $g(x) = \sqrt{x} + 2$.

1. **State the problem:** Amin wants to buy a water tank with a marked price of 32000 shillings and is given a 20% discount. 2. **Formula for discount price:** The price paid after

1. **State the problem:** Amina bought a pair of shoes for 2500 shillings and sold them at a 20% loss. We need to find the selling price. 2. **Formula used:** Selling Price (SP) wh

1. **Problem:** Find the domain of the function $$f(x) = \frac{1}{x^2 - 5x + 6}$$. 2. **Formula and rules:** The domain of a function with a denominator is all real numbers except

1. The problem is to simplify the expression $$-3x^2y^3(-1.1 - 2xy^2 + 0.5x - 2.3y^3)$$. 2. We use the distributive property: $$a(b + c + d + e) = ab + ac + ad + ae$$.

1. **State the problem:** We are given the function $$V(t) = 10 + \frac{20t + 5}{\sqrt{4t^2 + 9}}$$ which represents voltage as a function of time $t$. We want to understand and an

1. **Problem:** Difference of two numbers is 18 and their ratio is 10. Find the numbers. 2. **Step 1: Define variables.** Let the two numbers be $x$ and $y$ such that $x > y$.

1. **Problem:** Find the slope of a line parallel to the line $y = 3x + 2$. 2. **Formula and rule:** Lines that are parallel have the same slope.

1. **State the problem:** Solve the equation $7x + 3y - 3z = -9$ for one variable in terms of the others. 2. **Choose a variable to solve for:** Let's solve for $x$.

1. **Problem:** The ratio of first and second class fares between two stations is 5:3, and the ratio of passengers traveling in first and second class is 2:11. Total fare collected

1. **State the problem:** We have three functions:

1. مسئله: نوع عدد اعشاری مربوط به کسر 5/42 را مشخص کنید. 2. برای تعیین نوع عدد اعشاری کسر، باید مخرج کسر را تجزیه کنیم و ببینیم فقط شامل عوامل 2 و 5 است یا خیر.

1. **State the problem:** Simplify the expression $\frac{\sqrt{1}}{4 - x^2}$. 2. **Recall the properties:** The square root of 1 is 1 because $\sqrt{1} = 1$.

1. **State the problem:** Simplify the expression $$\frac{\sqrt[3]{2\sqrt{8}}}{\sqrt[3]{\sqrt{2\sqrt{2}}}}$$. 2. **Recall the rules:**

1. **Problem statement:** Find (i) the domain and range of functions $f(x) = \sqrt{\frac{1}{4} - x^2}$ and $h(x) = \tan(x - 3)$.