Finding the Y-Intercept of a Rational Function

In the realm of mathematical functions, finding the y-intercept is crucial for understanding the behavior of functions such as rational functions. A rational function is a function defined as a ratio of two polynomials. To determine the y-intercept of a rational function, we evaluate the function when x equals zero (x 0). This value of y gives us the intercept on the y-axis. Let's delve into the details.

Setting up the Equation

A rational function is of the form:

f(x) Px / Qx

Where Px and Qx are polynomials. The y-intercept is found by substituting x 0 into the function:

f(0) P(0) / Q(0)

This substitution will give us the value of y at the point where the function intersects the y-axis. If the value of Q(0) is not zero, then f(0) will yield the y-intercept. However, if Q(0) 0, the function is undefined at that point, and there is no y-intercept.

Examples and Asymptotes

Consider an example where f(x) 2x - 1 / (2x 1).

Here, Px 2x - 1 and Rx 2x 1. For the y-intercept, we check if Rx 0 when x 0.

Rx(0) 2(0) 1 1

Since Rx(0) ≠ 0, the function has a y-intercept at:

y f(0) 2(0) - 1 / (2(0) 1) -1

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

Undefined Y-Intercepts and Asymptotes

Consider another rational function: f(x) 2x - 1 / x.

If we directly substitute x 0, we encounter a divide-by-zero situation, which is undefined:

f(0) 2(0) - 1 / 0 -1/0

To handle this, we simplify the rational function:

f(x) {2x - 1} / x 2 - 1}{x}

As x → 0, the term 2 becomes negligible compared to the infinitely large 1}{x}. Thus, the function approaches negative infinity:

as x → 0, f(x) → 2 - 1}{0} 2 - ∞ -∞

This indicates that at x 0, the function approaches negative infinity, with x 0 acting as an asymptote. In this case, the function never touches the y-axis but instead goes to negative infinity as x approaches zero.

Conclusion

Understanding the y-intercept of a rational function is essential for analyzing and graphing such functions. By evaluating the function at x 0, we can determine the y-intercept or identify the presence of asymptotes. Properly analyzing these characteristics provides a deeper insight into the nature and behavior of rational functions.

Keywords

rational function, y-intercept, x0