Question

Rational Function Does the graph of every rational function have a vertical asymptote? Explain.

Short answer

No, not all rational functions have vertical asymptotes. This depends on the specific form of the function, particularly whether the denominator polynomial becomes zero at a value of x where the numerator polynomial does not.

Step-by-step solution

Understand vertical asymptotes

A vertical asymptote of a function is a vertical line x = a where the function tends to either positive infinity or negative infinity as x approaches a. Typically in a rational function (a function of the form f(x) = P(x)/Q(x), where P(x) and Q(x) are polynomials), we would find vertical asymptotes at values of x for which the denominator Q(x) is zero (as division by zero is undefined), while the numerator P(x) is not zero.

Understand exceptions to vertical asymptotes

However, there is an exception to this rule. If the denominator and the numerator both become zero at the same value of x, it can result in one of the two circumstances. If the order of the zero in the denominator is greater, then we still get a vertical asymptote. However, if the zeroes are of the same order, then that x value is a hole in the function rather than an asymptote.

Conclusion

So, not all rational functions have vertical asymptotes. Whether or not a given rational function has a vertical asymptote depends on the specific form of the function, particularly the properties of the denominator polynomial in relation to the numerator polynomial.