91Ó°ÊÓ

A rational function is a fraction where both the numerator and the denominator are polynomials. In our example, we have the function \[y = \frac{-1}{x+2}\]. Here, the numerator is \(-1\), and the denominator is the linear polynomial \(x + 2\). Rational functions can exhibit interesting behavior, like vertical asymptotes or horizontal asymptotes. For \[y = \frac{-1}{x+2}\], we see that the function will have a vertical asymptote at \(x = -2\) because the denominator becomes zero at this point. This causes the function to go to infinity or negative infinity. As \(x\) gets very large positively or negatively, the function value will approach zero, giving us a horizontal asymptote at \(y = 0\). These characteristics are essential when graphing or analyzing the function.

The horizontal line test is a quick way to check if a function is one-to-one. A function is one-to-one if every horizontal line drawn through its graph crosses it at most once. For our example function \[y = \frac{-1}{x+2}\], we need to analyze its graph. Notice that the function is strictly decreasing, meaning as \(x\) increases, \(y\) decreases. This implies that no horizontal line will cross the graph more than once at any point.
This behavior indicates that each \(y\) value maps to exactly one \(x\) value. Therefore, the function passes the horizontal line test and is one-to-one.
Understanding the horizontal line test can help quickly identify whether the one-to-one property holds for various functions. Remember that one-to-one functions are essential in many mathematical contexts, including inverse functions.

Analyzing a function involves understanding its behavior and characteristics. For \[y = \frac{-1}{x+2}\], we start by noting that it's a rational function. We see vertical asymptotes where the denominator is zero (e.g. \(x = -2\)). We also have horizontal asymptotes where the value of \(y\) steadies as \(x\) goes to infinity (here, \(y = 0\)).
We use the horizontal line test to ascertain if the function is one-to-one. For this function, since it decreases throughout its domain, it means no horizontal line can intersect the graph at more than one point.
Examining these features helps with sketching the graph and visually confirming the one-to-one nature of the function: it's strictly decreasing, so it meets the criteria for one-to-one functions.
In summary: by analyzing the rational function's asymptotes, behavior, and applying the Horizontal Line Test, we can comfortably claim the function is correctly identified as one-to-one.