Regardless of the type of rational function, if the X-intercept does not exist, then the horizontal asymptote is y=0. True or False?
Well, if an X-intercept does not exist for a rational function, it means that the graph never crosses the X-axis. In that case, the horizontal asymptote could still be at y=0, but it's not guaranteed.
However, just to keep things humorous, let's imagine a rebellious rational function that refuses to conform to the norm. This quirky function might say, "Oh, you think I don't have an X-intercept? Well, I won't have a horizontal asymptote at y=0 either! I'll just go off and do my own thing!" So, in short, the statement is False!
False.
The statement is not always true. The presence or absence of an x-intercept does not determine the horizontal asymptote of a rational function. The horizontal asymptote depends on the degrees of the numerator and denominator of the rational function.
If the degree of the numerator (let's say n) is less than the degree of the denominator (let's say m), then the horizontal asymptote is y = 0. However, if the degree of the numerator is equal to or greater than the degree of the denominator, the rational function may have a different horizontal asymptote.
For example, if the rational function has a higher degree in the numerator compared to the denominator, it may have a slant asymptote or no horizontal asymptote at all.
Therefore, the correct statement would be that if the degree of the numerator is less than the degree of the denominator, then the horizontal asymptote is y = 0.
False.
To determine the horizontal asymptote of a rational function, we need to examine the degrees of the numerator and denominator polynomials.
1. If the degree of the numerator polynomial is less than the degree of the denominator polynomial, then the horizontal asymptote is y = 0 (the x-axis).
2. If the degree of the numerator polynomial is equal to the degree of the denominator polynomial, then the horizontal asymptote is the ratio of the leading coefficients of the numerator and denominator.
3. If the degree of the numerator polynomial is greater than the degree of the denominator polynomial, then there is no horizontal asymptote. In this case, the function may exhibit a slant asymptote.
The existence of the X-intercept does not affect the determination of the horizontal asymptote.