Which of the following functions have a graph that does not have a vertical asymptote?
Select all that apply.
A. $f(x)=\frac{3}{x^{2}}$
B. $f(x)=\frac{1}{x^{2}-3}$
c. $f(x)=\frac{1}{x^{2}+3}$
D. $f(x)=\frac{3 x+1}{x-7}$

Step 1 ：A vertical asymptote occurs when the denominator of a function is equal to zero. Therefore, we need to find the values of x for which the denominator of each function is equal to zero. If there are no such values, then the function does not have a vertical asymptote.

Step 2 ：For function A, \(f(x)=\frac{3}{x^{2}}\), the denominator is zero when x = 0. Therefore, function A has a vertical asymptote at x = 0.

Step 3 ：For function B, \(f(x)=\frac{1}{x^{2}-3}\), the denominator is zero when x = \(-\sqrt{3}\) and x = \(\sqrt{3}\). Therefore, function B has vertical asymptotes at x = \(-\sqrt{3}\) and x = \(\sqrt{3}\).

Step 4 ：For function C, \(f(x)=\frac{1}{x^{2}+3}\), the denominator is never zero for real values of x. Therefore, function C does not have a vertical asymptote.

Step 5 ：For function D, \(f(x)=\frac{3 x+1}{x-7}\), the denominator is zero when x = 7. Therefore, function D has a vertical asymptote at x = 7.

Step 6 ：Thus, the only function that does not have a vertical asymptote is \(f(x)=\frac{1}{x^{2}+3}\).

Step 7 ：Final Answer: \(\boxed{f(x)=\frac{1}{x^{2}+3}}\)