Questions: Determine whether the following statement is true or false: The graph of a rational function may intersect a horizontal asymptote.

Transcript text: Determine whether the following statement is true or false: The graph of a rational function may intersect a horizontal asymptote.

Solution

Solution Steps

Step 1: Understanding Horizontal Asymptotes

A horizontal asymptote of a rational function \( f(x) = \frac{P(x)}{Q(x)} \) is a horizontal line \( y = L \) that the graph of the function approaches as \( x \) approaches \( \infty \) or \( -\infty \). The existence and location of horizontal asymptotes depend on the degrees of the numerator \( P(x) \) and the denominator \( Q(x) \).

Step 2: Analyzing Intersection with Horizontal Asymptotes

For a rational function, the graph can intersect a horizontal asymptote at specific points. This occurs when the function \( f(x) \) equals the value of the horizontal asymptote \( L \) for some finite \( x \). For example, if \( f(x) = \frac{x^2 + 1}{x^2 - 1} \), the horizontal asymptote is \( y = 1 \), and the graph intersects this asymptote at \( x = 0 \).

Step 3: Conclusion

Since the graph of a rational function can intersect its horizontal asymptote, the statement is True.