Question

Докажите, что функция y=x^3+x^2-6x/x^2-2x является линейной

Answer 1

(x³ + x² - 6x)/(x² - 2x) = (x(x² + x - 6))/(x(x - 2) = (x² + x - 6))/(x - 2)
Решаем квадратное уравнение x² + x - 6 = 0    D=b²-4ac=25=5²
                                                 x₁=(-b+√D)/2a=(-1+5)/2 = 2
                                                 x₂=(-b-√D)/2a=(-1-5)/2 = -3  тогда:
 (x² + x - 6))/(x - 2) = ((x - 2)(x + 3))/(x - 2) = x + 3

y = x + 3 - линейная функция, значит и исходная функция
у = (x³ + x² - 6x)/(x² - 2x) также является линейной

Допишу, чтобы понятнее было..)))
Любое квадратное уравнение вида  ax²+bx+c преобразуется в произведение вида:
a(x-x₁)(x-x₂), где x₁ и x₂ - корни данного квадратного уравнения

Answer 2

Checking if the Function is Linear

To determine if the function \( y = \frac{x^3 + x^2 - 6x}{x^2 - 2x} \) is linear, we need to consider the definition of a linear function.

A function is linear if it satisfies two properties: 1. It is of the form \( y = mx + b \), where \( m \) and \( b \) are constants. 2. It has a constant rate of change, meaning that the change in \( y \) is directly proportional to the change in \( x \).

Let's analyze the given function to see if it meets these criteria.

Analysis of the Given Function

The given function is \( y = \frac{x^3 + x^2 - 6x}{x^2 - 2x} \).

1. Form of the Function: - The given function is not in the form \( y = mx + b \), which is the standard form for a linear function. Instead, it is a rational function with a polynomial in the numerator and denominator. Therefore, it does not match the form of a linear function.

2. Rate of Change: - In a linear function, the rate of change is constant. This means that for every unit change in \( x \), there is a constant change in \( y \). However, in the given function, the rate of change is not constant due to the presence of the \( x^3 \) and \( x^2 \) terms in the numerator, and the \( x^2 \) and \( x \) terms in the denominator.

Conclusion

Based on the analysis, it is clear that the given function \( y = \frac{x^3 + x^2 - 6x}{x^2 - 2x} \) does not represent a linear function. It does not satisfy the standard form of a linear function, nor does it exhibit a constant rate of change.

Therefore, the function \( y = \frac{x^3 + x^2 - 6x}{x^2 - 2x} \) is not linear.

I hope this explanation helps! If you have further questions or need additional clarification, feel free to ask.