Question

Исследуйте функцию с графиком : y=2x/x^2+1

Answer 1

Вложение..........................................................................................................

Answer 2

Function Analysis: y = 2x / (x^2 + 1)

To analyze the function y = 2x / (x^2 + 1), we can examine its properties, such as the domain, range, symmetry, and behavior.

Domain: The domain of a function refers to the set of all possible input values (x-values) for which the function is defined. In this case, since the denominator x^2 + 1 is always positive, the function is defined for all real numbers. Therefore, the domain of y = 2x / (x^2 + 1) is all real numbers.

Range: The range of a function refers to the set of all possible output values (y-values) that the function can produce. To determine the range, we need to consider the behavior of the function as x approaches positive and negative infinity. As x approaches positive or negative infinity, the function approaches 0. Therefore, the range of y = 2x / (x^2 + 1) is all real numbers except 0.

Symmetry: To determine if the function has any symmetry, we can check if it is an even or odd function. An even function is symmetric with respect to the y-axis, meaning that f(x) = f(-x) for all x in the domain. An odd function is symmetric with respect to the origin, meaning that f(x) = -f(-x) for all x in the domain.

For the function y = 2x / (x^2 + 1), we can observe that f(x) = f(-x) does not hold, and f(x) = -f(-x) also does not hold. Therefore, the function does not exhibit any symmetry.

Behavior: To understand the behavior of the function, we can examine its limits as x approaches positive and negative infinity. As x approaches positive or negative infinity, the function approaches 0. This indicates that the function has horizontal asymptotes at y = 0.

Additionally, we can find the x-intercept(s) by setting y = 0 and solving for x. In this case, setting 2x / (x^2 + 1) = 0 gives us 2x = 0, which implies x = 0. Therefore, the function has an x-intercept at x = 0.

Graph of y = 2x / (x^2 + 1)

To visualize the graph of y = 2x / (x^2 + 1), we can plot some points and observe the overall shape of the function.

| x | y | |------|-------------------| | -3 | -0.153846153846... | | -2 | -0.4 | | -1 | -1 | | 0 | 0 | | 1 | 1 | | 2 | 0.4 | | 3 | 0.153846153846... |

By plotting these points and connecting them, we can see that the graph of y = 2x / (x^2 + 1) resembles a hyperbola. The function approaches the x-axis as x approaches positive or negative infinity, and it has a vertical asymptote at x = 0.


Please note that the graph is a rough approximation and may not be perfectly accurate.

I hope this helps! Let me know if you have any further questions.