Understanding the X-Intercepts of the Function y x - 2x - 1x3

The x-intercepts of a function are the points where the function crosses the x-axis. This means that at these points, the y-value is zero. For the function y x - 2x - 1x3, the x-intercepts can be found by setting the y-value equal to zero and solving for x. Here, we will go through the process step-by-step and present the solutions in a clear and detailed manner.

Setting the y-Value to Zero and Solving for x

To find the x-intercepts of the function (y x - 2x - 1x^3), we start by setting the equation to zero:

(0 x - 2x - 1x^3)

By rearranging this equation, we can express it as a product of factors:

(x - 2x - 1x^3 0)

This can be rewritten as:

((x - 2)(x - 1)(x 3) 0)

Finding the Values of x That Satisfy the Equation

The equation ((x - 2)(x - 1)(x 3) 0) is a product of three factors that equals zero. According to the zero-product property, if a product of factors is zero, then at least one of the factors must be zero. Therefore, we can set each factor equal to zero and solve for x:

Finding Each Root or Zero

Setting (x - 2 0) we find:

x  2

Setting (x - 1 0) we find:

x  1

Setting (x 3 0) we find:

x  -3

Therefore, the x-intercepts of the function are ((2, 0)), ((1, 0)), and ((-3, 0)).

Conclusion

In summary, the function (y x - 2x - 1x^3) has three x-intercepts: x 2, x 1, and x -3. These intercepts are also referred to as the roots or zeros of the function. The points where these intercepts occur on the coordinate plane are as follows:

Understanding these intercepts helps to visualize and analyze the behavior of the function, providing valuable insights for further mathematical exploration and real-world applications.