🧮 algebra

1. **State the problem:** We want to study the function $$f(x) = \sqrt{\frac{1+\sin x}{1-\sin x}}$$. 2. **Recall important formulas and rules:**

1. **State the problem:** We are given the functional equation $$f\left(\frac{x-1}{x}\right) = f(x) - x^2$$ and need to find the function $f(x)$.\n\n2. **Analyze the equation:** Th

1. **Stating the problem:** Simplify or analyze the expression $$\frac{x}{y+x}$$ and understand the given equation $$\frac{x}{y+x} = \frac{1}{y}$$. 2. **Understanding the equation:

1. Problem: Odrediti parametar $k$ tako da rješenja jednadžbe $$k^2x^2 - k(5k+1)x - (4k+2) = 0$$ budu kompleksna. 2. Za kvadratnu jednadžbu oblika $$ax^2 + bx + c = 0$$, rješenja s

1. The problem is to simplify the expression "Simplified" which appears to be incomplete or missing specific terms. 2. To simplify an expression, we combine like terms, apply arith

1. **State the problem:** We have three airplanes A, B, and C flying along paths represented by lines on a coordinate plane. - Airplane A's path passes through points $(2,-2)$ and

1. The problem states: Given the function $f(x) = 7x^3 + 5x - 6$, find $\frac{4}{f(x)}$. 2. The function $f(x)$ is a cubic polynomial: $f(x) = 7x^3 + 5x - 6$.

1. **State the problem:** We are given the linear equation $$2x - 3y + 24 = 0$$ and several points. We want to check which points lie on this line.

1. **Problem statement:** Find the domain of the function $$y = \sqrt{4x - 3} + \frac{1}{6 - x}.$$\n\n2. **Recall domain rules:**\n- The expression under the square root must be no

1. مسئله: دامنه تابع $f(x) = -3\tan(3x) + 1$ را بیابید. تابع تانژانت در نقاطی که مخرج کسر در تعریف تانژانت صفر می‌شود، تعریف نشده است. تانژانت به صورت $\tan(\theta) = \frac{\sin(\t

1. Diberikan ekspresi $$\frac{64^{\frac{3}{4}} + 81^{\frac{1}{2}}}{125^{\frac{3}{5}} - 32^{\frac{1}{5}}}$$. Kita diminta mencari nilai paling sederhana dari ekspresi ini. 2. Gunaka

1. **Problem statement:** Given a population growth scenario where the population increases to 11000, find the number of years it takes for this increase. 2. **Formula used:** The

1. **State the problem:** We need to multiply and simplify the expression for total weight growth per basket given by $$W(x) = D(x) \times R(x)$$ where $$D(x) = \frac{4}{x+2}$$ and

1. The problem asks to simplify $\sqrt{32}$ and enter the simplified numerical coefficient only. 2. Recall the property of square roots: $\sqrt{a \times b} = \sqrt{a} \times \sqrt{

1. The problem asks us to find the values of $x$ that make the rational expression $$\frac{4}{x(x - 3)}$$ undefined. 2. A rational expression is undefined when its denominator equa

1. **State the problem:** Multiply the expressions $$\frac{x}{x+3} \cdot \frac{x+3}{4}$$. 2. **Recall the multiplication rule for fractions:** When multiplying fractions, multiply

1. The problem asks for the domain of the function $f(x) = \sqrt{x - 4}$.\n\n2. The domain of a square root function is all values of $x$ for which the expression inside the square

1. **Factorise the expression 9p^2 - q^2** This is a difference of squares, which follows the formula:

1. **State the problem:** We need to find a rational expression for the total amount of food per fasting person, given: - Rice per person: $\frac{2}{x}$ kilograms

1. The problem asks us to analyze the behavior of the graphs of the functions $f(x) = x^3 - 1$ and $g(x) = -x^3 + 1$ for large values of $x$. 2. The key idea is to understand the e

1. **State the problem:** Find real numbers $a$ and $b$ such that $a + bi = -9 + 4i$. 2. **Recall the rule:** Two complex numbers are equal if and only if their real parts are equa