🧮 algebra

1. **Problem statement:** Given functions $f(x) = x + 2$ and $g(x) = x^2 - 9$, find $g(f(x))$. 2. **Formula and rules:** To find $g(f(x))$, we substitute $f(x)$ into $g(x)$, i.e.,

1. The problem asks for the coefficient of $x^3$ in the expansion of $(1 + x)^5$. 2. We use the Binomial Theorem formula for expansion: $$(a + b)^n = \sum_{k=0}^n \binom{n}{k} a^{n

1. مسئله: تعداد جواب‌های معادله $$\sqrt{x-1} + \sqrt{x-4} = \sqrt{x-5}$$ را بیابید. 2. دامنه تعریف: برای رادیکال‌ها باید زیر رادیکال‌ها غیرمنفی باشند:

1. **State the problem:** We are given a function $h(t)$ defined by the table: | t | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |

1. The problem asks for the sum of the infinite geometric series: $$1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots$$ 2. The formula for the sum $S$ of an infinite geometric se

1. **Problem statement:** Given the function values from the table: $$\begin{array}{c|ccccccccc}

1. **Problem:** Find a simplified form for the term $t_n$ of the sequence 5, 13, 21, ... 2. **Formula:** For an arithmetic sequence, the $n$th term is given by

1. The problem asks for the 4th term of a geometric sequence with first term $a_1=2$ and common ratio $r=3$. 2. The formula for the $n$th term of a geometric sequence is:

1. The problem asks for the sum of the first 10 terms of the arithmetic sequence $2, 5, 8, 11, \ldots$. 2. The formula for the sum of the first $n$ terms of an arithmetic sequence

1. **Problem statement:** Find the area of a park with width $\sqrt{15}$ meters and length $7\sqrt{21}$ meters. 2. **Formula:** The area $A$ of a rectangle is given by

1. **Problem:** Expand and simplify the expression $\left(2\sqrt{12} - \sqrt{5}\right)\left(\sqrt{8} - \sqrt{10}\right)$. 2. **Formula and rules:** Use the distributive property (F

1. **Problem 2.1:** Find the domain and range of the function $h(x) = \sqrt{x^2 - 9}$.\n\n2. **Formula and rules:** The square root function $\sqrt{y}$ is defined only when $y \geq

1. The problem asks to find the value of the function $f(x) = 2x - 3$ at $x=1$. 2. The formula for evaluating a function at a given input is to substitute the input value into the

1. The problem asks: What percentage of the total money shared does Miriam receive? 2. To find the percentage of total money received by Miriam, use the formula:

1. **Stating the problem:** Some money is shared between Mariam and Nina in the ratio of 2 : 3. 2. **Understanding the ratio:** The ratio 2 : 3 means for every 2 parts Mariam gets,

1. Stated problem: Evaluate the expression $$\sqrt[3]{1331} + 16 \div \frac{1}{3} - 21 - (-29) - \sqrt{16} \times 2 - 35 + \frac{5^2}{8} \times 6 \times \frac{1}{4} + 37 - 2^5 \tim

1. Diberikan ekspresi dengan akar kubik \(\sqrt[3]{780}\).\n2. Karena 780 bukan kubus sempurna, kita biarkan dalam bentuk akar kubik untuk hasil eksak, atau gunakan aproksimasi \(\

1. Diberikan garis awal $y = ax + b$. 2. Garis digeser 4 satuan ke kanan dan 5 satuan ke bawah, sehingga persamaan garis menjadi:

1. **State the problem:** Evaluate the expression $2+4(7)$. 2. **Recall the order of operations:** According to PEMDAS/BODMAS, multiplication is performed before addition.

1. **Problem Statement:** Sketch the graph of the function $$f(x) = (x - 1)^2 (x + 1)^3$$. 2. **Formula and Important Rules:** The function is a product of two factors: a squared t

1. **State the problem:** We need to sketch the graph of the quadratic function $$f(x) = 2x^2 - 4x - 2$$. 2. **Recall the standard form and vertex formula:** A quadratic function i