Question

give an example of a function f(x) such that 1/f(x) has vertical asymptotes at x=4 and x=-3

Answer 1

Vertical asymptotes are the result of division by 0, so essentially what you want who evaluates to 0 (has roots) at x=4 and x=-3

Answer 2

To have a function with roots \(r_1, r_2, \dots , r_n\), you basically do this: \[
f(x) = (x-r_1)(x-r_2)...(x-r_n)
\]Notice how if \(x=r_1\)\[
f(x) = (r_1-r_1)(x-r_2)...(x-r_n) =(0)(x-r_2)...(x-r_n) = 0
\]Since just one \(0\) makes the whole thing \(0\), \(x\) can be equal to any root and \(f(x)=0\).

Answer 3

In our case, \(r_1 = 4,r_2=-1\), which gives us: \[
f(x) = (x-(4))(x-(-3)) = (x-4)(x+3) = x^2-x-12
\]