🧮 algebra

1. The problem is to factor the quadratic expression $x^2 + 5x + 6$. 2. We look for two numbers that multiply to the constant term 6 and add up to the coefficient of $x$, which is

1. **State the problem:** Solve the equation $$\frac{x}{-3} + 5 = 10$$ for $x$. 2. **Isolate the term with $x$:** Subtract 5 from both sides:

1. The problem appears to involve the expression $\frac{a}{b}$ followed by multiple mapping arrows $\mapsto\mapsto\mapsto$. 2. Typically, $\frac{a}{b}$ represents a fraction or rat

1. **State the problem:** Factor the quadratic expression $x^2 + 5x + 6$. 2. **Identify coefficients:** The quadratic is in the form $ax^2 + bx + c$ where $a=1$, $b=5$, and $c=6$.

1. **State the problem:** We have an increasing function $f$ on its domain. We want to determine which of the given functions is not necessarily increasing on its domain. 2. **Anal

1. The problem states that the graph of the function $g(x) = mx^2$ is a parabola with vertex at the origin $(0,0)$ and passes through the point $(-2, 20)$. We need to find the valu

1. The problem asks us to determine the interval(s) where the function $f(x) = -4$ is negative. 2. The function $f(x) = -4$ is a constant function, meaning it has the same value fo

1. **State the problem:** Factor the quadratic expression $x^2 + 5x + 6$. 2. **Identify coefficients:** The quadratic is in the form $ax^2 + bx + c$ where $a=1$, $b=5$, and $c=6$.

1. **Problem 5.4:** Show that \(\left(3 - \frac{1}{\sqrt{5}}\right) \left(9 + \frac{3}{\sqrt{5}} + \frac{1}{5}\right)\) can be expressed as \(a - \frac{1}{25} \sqrt{b}\) where \(a,

1. **State the problem:** (a) Two variables $x$ and $y$ are such that the rate of change of $y$ with respect to $x$ is proportional to $y$. We need to state a model for $y$ in term

1. The problem is to solve the equation $5.5$. 2. Since $5.5$ is a constant number and not an equation, it is already simplified.

1. **State the problem:** We are given the function $$y=1-2e^{\frac{1}{2}x+1}$$ and asked to identify the basic function, sketch it over the interval $$0 \leq x \leq 1$$, identify

1. **State the problem:** Find the domain of the function $$k(x) = \frac{1}{\sqrt{4 - x}}$$. 2. **Analyze the function:** The function involves a square root in the denominator. Fo

1. **State the problem:** Show that \( (3 - \frac{1}{\sqrt{5}})(9 + \frac{3}{\sqrt{5}} + \frac{1}{5}) \) can be expressed as \( a - \frac{1}{25} \sqrt{b} \) where \( a, b \in \math

1. **נתון:** הפונקציה $$f(x)=\frac{e^{2x} - 9e^x}{e^{2x} - 10e^x + 9}$$. 2. **מציאת תחום ההגדרה:**

1. נתחיל בהגדרה כללית: משוואה היא ביטוי מתמטי שמכיל שני צדדים שמחוברים באמצעות סימן שוויון $=$. 2. מטרת פתרון המשוואה היא למצוא את הערך או הערכים של המשתנה שמקיימים את השוויון.

1. **State the problem:** Solve the equation $$9^x + 9 = 10 \cdot 3^x$$ for $x$. 2. **Rewrite the bases:** Note that $9 = 3^2$, so rewrite $9^x$ as $(3^2)^x = 3^{2x}$.

1. **נתונה הפונקציה:** $$f(x) = \frac{e^{2x} - 9e^x}{e^{2x} - 10e^x + 9}$$

1. Solve $5^{-x} \cdot 5^{x-2} = \frac{25^{2x}}{5}$ Combine powers on the left: $5^{-x + x - 2} = 5^{-2}$

1. **State the problem:** We are given the function $$f(x) = \frac{e^{2x} - 9e^x}{e^{2x} - 10e^x + 9}$$ and want to analyze its behavior. 2. **Simplify the expression:** Let $$y =

1. Problem 10: Given $f(x)= \frac{x}{3x+2}$ and $g(x) = -\frac{4}{x}$, find $g \circ f$ and its domain. 2. To find $g(f(x))$, substitute $f(x)$ into $g$: