🧮 algebra

1. **State the problem:** We have a sequence where the term number $t$ and term value $v$ are given as follows: $t=1, v=11$; $t=2, v=22$; $t=3, v=33$; $t=4, v=44$. We need to descr

1. **State the problem:** We want to find an equation relating the number of squares $n$ in the pattern to the size number $s$. 2. **Analyze the pattern:**

1. **State the problem:** We have the function $f(x) = \log_{\frac{1}{3}} x$ with point $A$ as the x-intercept and point $(3, t)$ on the graph. We need to find $t$, coordinates of

1. The problem asks to evaluate the ceiling and floor functions for given values. 2. For part a, \( \lceil 49.4 \rceil \) means the smallest integer greater than or equal to 49.4.

1. **State the problems:** We have two separate problems to solve:

1. **State the problem:** We have a torpedo depth modeled by a quadratic pattern with depths at 1, 2, and 3 seconds given. We need to find the depth at 5 seconds, verify the formul

1. **State the problem:** We are given two functions $f(x) = 3x - 2$ and $g(x) = 2x + k$. We need to find the value of $k$ such that the composition $f \circ g = g \circ f$. 2. **W

1. Solve $(x+5)(x-2)=0$. Use the zero product property: $x+5=0$ or $x-2=0$.

1. The problem asks to find the compositions $f \circ g$ and $g \circ f$ where $f(x) = 2x + 1$ and $g(x) = x^2 - 2$. 2. First, find $f \circ g$, which means $f(g(x))$.

1. **State the problem:** We are given the function $$S(t) = \frac{1}{2}gt^2 + at + b$$ where $g$ is the acceleration due to gravity, and $a$, $b$ are constants. We need to check i

1. **State the problem:** We need to show that the function $f: \mathbb{R} \setminus \{0\} \to \mathbb{R} \setminus \{0\}$ defined by $f(x) = \frac{1}{x}$ is a bijection. 2. **Show

1. The problem asks to find the domain and range of the function $f(x) = x^2 + 1$. 2. The domain of a function is the set of all possible input values ($x$) for which the function

1. **Stating the problem:** Solve the equation $$\frac{1\frac{1}{2}p}{2} - 1 = \frac{3x}{3} + 6$$. 2. **Rewrite mixed number:** Convert $1\frac{1}{2}$ to an improper fraction: $$1\

1. **State the problem:** Find all unordered pairs of integers $(x,y)$ such that $$x^2 - y^2 = 75.$$ 2. **Rewrite using difference of squares:** We factor the left side as $$x^2 -

1. **State the problem:** Find the equation of the normal to the circle given by $$x^2 + y^2 - 2x - 4y - 12 = 0$$ at the point $$(-3, 5)$$. 2. **Rewrite the circle equation in stan

1. The problem is to find all unordered pairs of integers $(x,y)$ such that $$x^2 - y^2 = 75.$$\n\n2. Rewrite the equation using the difference of squares factorization: $$x^2 - y^

1. The problem is to find the number of unordered pairs $(x,y)$ where both $x$ and $y$ are positive and satisfy the equation $$x^2 - y^2 = 75.$$\n\n2. Rewrite the equation using th

1. The problem is to find the number of positive integer pairs $(x,y)$ such that $x^2 - y^2 = 49$. 2. Rewrite the equation using the difference of squares factorization:

1. The problem is to find the number of unordered pairs $(x,y)$ where $x$ and $y$ are integers satisfying the equation $$x^2 - y^2 = 49.$$ 2. Rewrite the equation using the differe

1. The problem is to analyze the equation of the form $$x^2 - y^2 = 49$$. 2. This equation represents a hyperbola because it is in the form $$x^2/a^2 - y^2/b^2 = 1$$ where $$a^2 =

1. **State the problem:** Five people went to a restaurant. Four of them each spent $12 less than the average amount spent by all five together, and the fifth person spent $18 less