🧮 algebra

1. **Problem Statement:** Determine if each function $g: \mathbb{R} \to \mathbb{R}$ is one-to-one (injective) and onto (surjective). If not onto, find the range $g(\mathbb{R})$. 2.

1. **State the problem:** We need to find the value of $r$ such that $(y+2)$ is a factor of the polynomial $3y^2 - 4ry - 4r^2$. 2. **Recall the factor theorem:** If $(y+2)$ is a fa

1. **State the problem:** We want to determine if it is possible to walk 2595 miles in 6 months. 2. **Identify the formula:** To find out if this is possible, we calculate the aver

1. Let's state a similar problem: Solve for $x$ in the equation $$2x + 5 = 15$$. 2. The formula used here is to isolate $x$ by performing inverse operations. Important rules: addit

1. The problem is to simplify and solve the given equation involving variables $y$, $a$, $b$, and $x$. 2. The equation appears to be: $$y + ax + yb + \varepsilon\varepsilon + y + r

1. **State the problem:** We have two numbers, say $a$ and $b$, and their product is $ab$. 2. **What is asked:** We want to find the increase in the product when the first number $

1. Let's start by understanding that you want an explanation suitable for an 8th class student. 2. When solving math problems, it's important to break them down into simple steps.

1. **Problem statement:** We are given two numbers whose product is $ab$. We want to find the increase in the product when the first number is increased by 4. 2. **Initial product:

1. **State the problem:** We are given two numbers, say $x$ and $y$, with the product $xy = ab$. We want to find the increase in the product when the first number $x$ is increased

1. **Problem:** Given the equation $\frac{x - 2y}{2 + 3y} = \frac{1}{3}$, find $\frac{y}{x}$. Also given are the equations $x - 2y = 3$ and $x + 3y = 1$. 2. **Step 1:** Use the giv

1. **Problem:** If the product of two numbers is $ab$, find the increase in the product when the first number is increased by 4. 2. **Formula and Explanation:** The original produc

1. **State the problem:** Simplify the expression $$(1 - \sin x\theta)(1 - \sin x\theta)$$. 2. **Formula used:** When multiplying two identical binomials, use the formula for the s

1. **Problem statement:** Given the functions $f(x) = x^2 - 5x$ and $g(x) = x - 5$, find the degree of each function and prove that $f(5) = g(5) = 0$. 2. **Degree of a polynomial:*

1. **Stating the problem:** We are given the equation $3^x = 5^y = (75)^z$ and need to show that $xy = z(y + 2x)$. 2. **Rewrite the equation:** Since $3^x = 5^y = (75)^z$, let this

1. **State the problem:** A car and a truck start from two towns 550 km apart, traveling towards each other. The car travels at 72 km/h and the truck at 38 km/h. We need to find th

1. **State the problem:** Find Farhan's average speed for the entire triathlon, given three segments: swimming, cycling, and running. 2. **Given data:**

1. **State the problem:** Given the arithmetic progression (AP) $x - 6, x - 1, x + 4, x + 9, \ldots$, find: (a) the first term $a$ and common difference $d$

1. **Stating the problem:** We need to find the annual income of Janathika Maratha given the tax slabs and the total tax paid of 15000 rupees.

1. **Problem statement:** Determine whether the function is even, odd, or neither, and find the zeroes of the function for part (a): $f(x) = 2x^3 - 4x$. 2. **Recall definitions:**

1. The problem asks to draw the graph of the relation $A, R = \{(x,y) : y = 2x\}$. 2. This is a linear equation representing a straight line with slope 2 passing through the origin

1. **State the problem:** Simplify the expression $3y - 6y + 3$. 2. **Identify like terms:** The terms $3y$ and $-6y$ are like terms because they both contain the variable $y$.