Understanding the Y-Intercept of a Rational Function

The y-intercept of a function is the point where the graph of the function intersects the y-axis. This intersection point can be found by setting x 0 in the function's equation and solving for y.

Introduction to Rational Functions

A rational function is a function that can be expressed as the ratio of two polynomials, i.e., fx Px/Rx, where Px is the numerator polynomial and Rx is the denominator polynomial. The y-intercept is determined by substituting x 0 into the function's equation, provided that the denominator does not become zero.

Example with a Rational Function

Consider the rational function fx (2x - 1) / (2x 1).

Step 1: Substitute x 0 into the function to find the y-intercept.

fx(0) (2(0) - 1) / (2(0) 1) -1 / 1 -1

Step 2: Verify that the denominator is not zero.

2x 1 2(0) 1 1 ne; 0

Therefore, the y-intercept is at the point (0, -1).

Dealing with Asymptotes

If substituting x 0 into the function makes the denominator zero, it indicates the presence of an asymptote, and the function does not have a y-intercept at that point.

For example, consider the function fx 2x - 1 / x.

Step 1: Substitute x 0 into the function.

fx(0) (2(0) - 1) / 0 -1 / 0

This results in a division by zero, which is undefined. Therefore, there is no y-intercept in this case.

Step 2: Simplify the function to gain further insight.

fx (2x / x) - (1 / x) 2 - (1 / x)

As x rarr; 0, the term - (1 / x) approaches negative infinity, indicating that the function's value approaches negative infinity.

Therefore, the y-axis is a vertical asymptote, and the function does not cross the y-axis.

General Approach

To find the y-intercept of any function:

Set x 0 in the function's equation. Solve for y. Ensure that the denominator is not zero. If it is zero, the function has no y-intercept at that point.

This approach works for both polynomial and rational functions. For example:

fx x5 / (x - 2)
y-intercept: fx(0) 5 / (0 - 2) -5/2

The y-intercept is at the point (0, -5/2).

Conclusion

Finding the y-intercept of a rational function is a straightforward process as long as you follow the steps carefully. Remember to check if the denominator becomes zero when x 0. If so, the function has an asymptote at that point and no y-intercept.