Question

Постройте график функции y=(x^2-x)|x|\x-1 и определите,при каких значениях m прямая y=m не имеет с графиком ни одной общей точки.

Answer 1

Y=(x^2-x)|x|\x-1⇔y=x *lxl; y=m⇒m=x* lxl⇔m=x², при х≥0 и m=-x²,при х<0, т.е
-x²<m<x²

Answer 2

Graph of the Function y=(x^2-x)|x|/x-1

To begin, let's plot the graph of the function y=(x^2-x)|x|/x-1 and then determine the values of m for which the line y=m does not intersect the graph.

The function y=(x^2-x)|x|/x-1 can be broken down into two separate functions based on the absolute value of x. When x is positive, the function is y=(x^2-x)x/x-1, and when x is negative, the function is y=(x^2-x)(-x)/x-1.

Let's plot the graph of these two functions separately and then combine them to understand the overall graph.

Graph of y=(x^2-x)x/x-1 for x > 0

The function y=(x^2-x)x/x-1 can be simplified to y=x(x-1) for x > 0. This is a quadratic function, and its graph will be a parabola.

Graph of y=(x^2-x)(-x)/x-1 for x < 0

The function y=(x^2-x)(-x)/x-1 can be simplified to y=-x(x-1) for x < 0. This is also a quadratic function, and its graph will be a parabola, but it will open in the opposite direction compared to the previous one.

By combining the graphs of these two functions, we can visualize the overall graph of y=(x^2-x)|x|/x-1.

Determining Values of m for which the Line y=m Does Not Intersect the Graph

To find the values of m for which the line y=m does not intersect the graph, we need to consider the behavior of the graph of y=(x^2-x)|x|/x-1.

The line y=m will not intersect the graph of y=(x^2-x)|x|/x-1 if and only if the value of m is outside the range of the function's output. In other words, if m is either greater than the maximum value of the function or less than the minimum value of the function, the line y=m will not intersect the graph.

To determine the maximum and minimum values of the function, we can analyze the behavior of the parabolas and their vertex points.

Let's proceed with these calculations to determine the specific values of m for which the line y=m does not intersect the graph of y=(x^2-x)|x|/x-1.