0

# Explain why the graph of the rational function f(x)= (-1)/(X^2+4) has no vertical asymptotes

I believe it is because there are no x factors in the numerator but unsure.

### 2 Answers by Expert Tutors

Michael J. | Effective High School STEM Tutor & CUNY Math Peer LeaderEffective High School STEM Tutor & CUNY ...
5.0 5.0 (5 lesson ratings) (5)
0
If we set the denominator equal to zero and solve for x, we won't get a real solution.  Therefore, the graph does not have any vertical asymptotes.  Therefore, the function is continuous.

x2 + 4 = 0

x2 = -4

x = 2i    and   x = -2i

These solutions are complex.
Arturo O. | Experienced Physics Teacher for Physics TutoringExperienced Physics Teacher for Physics ...
5.0 5.0 (66 lesson ratings) (66)
0
In order to have vertical asymptotes, the denominator x2 + 4 would have to be zero for some value of x.  But x2 + 4 > 0 for all real values of x.  (There are complex values of x that make x2 + 4 = 0, but I assume we are concerned only with real numbers here.)