🧮 algebra

1. **Statement of the problem:** Study the function $$f(x) = \frac{x^3 - 2x^2}{(x - 1)^2}$$ defined on $$\mathbb{R} \setminus \{1\}$$ and examine limits, asymptotes, derivatives, t

1. **State the problem:** Calculate the value of $$\log_{\frac{1}{2}} 8 + \log_2 \frac{1}{8} + \log_3 \frac{1}{9}.$$

1. Stating the problem: We analyze the set $E = \{-\sqrt{3}; \frac{22}{7}; \sqrt{\frac{12}{27}}; \frac{\pi}{3}; \sqrt{0.90}; 0.12131415\}$ and points $A(3,2), M(3,2), N(-3,0)$, wit

1. We are asked to analyze the expression $$e^{\sqrt{2x}+2}$$. 2. This is an exponential function with base $e$ (Euler's number) raised to the power of the sum of $\sqrt{2x}$ and 2

1. The problem is to compute the values of various quadratic functions at integer points from $x = -4$ to $x = 4$. 2. Each function is of the form $f(x) = ax^2 + c$, where $a$ and

1. **Problem a:** Find the domain, intercepts, range, and asymptotes of $$f(x)=\frac{2x-1}{x^2-4}$$ and sketch its graph. 2. **Domain:** The denominator cannot be zero.

1. The user asked to graph three functions, but no functions were provided in the previous context. 2. To assist properly, I need the explicit functions to plot.

1. Stating the problem: Calculate the value of the expression $$4^{2020} + 2^{2017} - 15$$.

1. **Problem a: Analyze and graph** $f(x) = \frac{2x - 1}{x^2 - 4}$. 2. **Domain:** The function is undefined where the denominator is zero: solve $x^2 - 4 = 0$ which gives $x = \p

1. We need to find the value of the expression $$729^8 - 3^{18} + 85$$ and then convert it to base 9 to find the number of significant zeros in its representation. 2. Note that $$7

1. **State the problem:** We want to identify the shape of the graph of the function $$y = -2x^2 + 15x - 6$$. 2. **Analyze the function form:** This function is a quadratic functio

1. The problem is to find the maximum integer value of the function $$y = -4^{x+2} - 3$$. 2. Observe that the term $$4^{x+2}$$ is an exponential function with base 4, which is alwa

1. **State the problem:** Solve the inequality $$\log_{0.5}(x - 3) < 1$$ and determine which interval satisfies this inequality from the given options.

1. We are given the quadratic equation $$6x^2 - 15x - 7 = 0.$$ 2. The product of the roots of a quadratic equation $$ax^2 + bx + c = 0$$ is given by $$\frac{c}{a}$$.

1. **Stating the problem:** We want to analyze the function $$f(x) = \frac{\cos\left(\frac{2x}{3}\right) + \sin\left(\frac{3x}{2}\right)}{2 - \cos\left(\frac{x}{4}\right)}.$$\n\n2.

1. The user has provided a function notation $F(x) =$ but has not provided the explicit function definition. 2. To proceed, please provide the expression for $F(x)$ so that I can h

1. **State the problem:** Find the Highest Common Factor (HCF) of the numbers 36, 24, and 12. 2. **Prime factorize each number:**

1. We start with the expression: $$\frac{2x - 64}{12}$$ 2. Factor out the common factor of 2 in the numerator: $$\frac{2(x - 32)}{12}$$

1. a. Simplify the division of fractions $\frac{3ab}{2c} : \frac{9b^2}{4ac}$ by multiplying the first fraction by the reciprocal of the second: $$\frac{3ab}{2c} \times \frac{4ac}{9

1. Let's first identify the problem: you have a sign error in your calculations. 2. A sign error means that either a plus (+) should be minus (−), or vice versa, in some part of yo

1. Sederhanakan pembagian bentuk aljabar berikut: a. $\frac{3ab}{2c} : \frac{9b^2}{4ac}$