🧮 algebra

1. **Problemstellung:** Wir sollen die Koordinaten der Schnittpunkte zweier Parabeln berechnen. 2. **Formel und Vorgehen:** Schnittpunkte zweier Funktionen $y=f(x)$ und $y=g(x)$ fi

1. **State the problem:** We are given the function $f(x) = -4\sqrt{-x}$ and want to understand its properties and graph. 2. **Recall the square root function:** The basic square r

1. The problem asks to identify the type of function based on the graph description. 2. The graph starts near (0, -2), increases gradually, passes through (4, 0), and curves slight

1. **State the problem:** We need to find the value of $x$. 2. **Identify the equation:** Since the user did not provide an explicit equation, we assume the problem is to solve for

1. **State the problem:** Solve the inequality $$-2(-3x + 2) \neq 20$$. 2. **Apply the distributive property:** Multiply $$-2$$ by each term inside the parentheses.

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 to undo th

1. **State the problem:** Simplify the expression $$\left(\frac{a^{-1}}{-ab \cdot 2a^3}\right)^2$$. 2. **Rewrite the denominator:** The denominator is $$-ab \cdot 2a^3 = -2ab a^3$$

1. **State the problem:** We are given two functions $g(x) = 4x - 4$ and $f(x) = x + 4$. We need to find the function $\left(\frac{g}{f}\right)(x)$, which means dividing $g(x)$ by

1. **State the problem:** We are given two functions $f(x) = 4x + 3$ and $g(x) = x + 5$. We need to find the product function $(f \cdot g)(x)$, which means multiplying $f(x)$ and $

1. **State the problem:** Solve the equation $$\frac{2x+3}{4} = 5$$ for $x$. 2. **Formula and rules:** To solve for $x$ in a fraction equation, multiply both sides by the denominat

1. **Problem statement:** Find the compositions $[f \circ g](x)$ and $[g \circ f](x)$ for the functions: $f(x) = x + 5$ and $g(x) = x - 3$

1. **State the problem:** Solve the inequality $2 - 3a < -8$ for $a$. 2. **Isolate the term with $a$:** Subtract 2 from both sides:

1. **State the problem:** Solve the inequality $$-13 \geq 2(b - 6) + 3$$ for $b$. 2. **Apply the distributive property:**

1. Das Problem: Du möchtest verstehen, wie Quotientengleichheit und Produktgleichheit funktionieren. 2. Quotientengleichheit bedeutet, dass wenn zwei Brüche gleich sind, also $$\fr

1. **State the problem:** We need to find the cost per issue of a monthly magazine subscription that costs 15.96 for one year. 2. **Identify the formula:** Since the subscription i

1. **State the problem:** We need to factor the polynomial $$w^6 - 7w^3 + 10$$. 2. **Rewrite the polynomial:** Notice that $$w^6 = (w^3)^2$$, so let $$x = w^3$$. The polynomial bec

1. The problem asks to find the expression equivalent to $36x^2 - 169$. 2. Recognize that $36x^2 - 169$ is a difference of squares because $36x^2 = (6x)^2$ and $169 = 13^2$.

1. The problem asks to find which expression is equivalent to $50x^2 - 128$. 2. We start by factoring the expression $50x^2 - 128$.

1. The problem asks us to find an expression equivalent to $x^3 - 125$. 2. Recognize that $x^3 - 125$ is a difference of cubes since $125 = 5^3$.

1. The problem states the relationship $d = \sqrt{\frac{3h}{2}}$ where $d$ is a distance and $h$ is a height with $h \geq 0$. 2. This formula models how distance $d$ depends on hei

1. **State the problem:** We start with the parent function $f(x) = \sqrt[3]{x}$. We want to find the new function $g(x)$ after three transformations: - Reflection over the x-axis