Question

Which statement describes the graph of the function?
The function has holes when x = 0, x = 3, and x = 5.
The function has vertical asymptotes when x = 0, x = 3, and x = 5.
The function has holes when x = 0 and x = 3, and a vertical asymptote when x = 5.
The function has a hole when x = 3, and vertical asymptotes when x = 0 and x = 5.

Answer 1

I don’t see the function you’re asking about. To pick the correct statement you must factor the rational function and see which zeros of the denominator cancel with zeros of the numerator.

How to decide:
- Factor numerator and denominator.
- If (x − a) is a common factor and cancels, there is a hole at x = a (the y-value is the simplified function’s value at a).
- If (x − a) is a factor of the denominator that does NOT cancel, there is a vertical asymptote at x = a.

Examples that give each choice:
- Holes at x = 0, 3, 5: f(x) = (x(x−3)(x−5)(x+1)) / (x(x−3)(x−5)) → simplified = x+1 (holes at 0, 3, 5).
- Vertical asymptotes at 0, 3, 5: f(x) = 1 / (x(x−3)(x−5)) (no cancelling factors).
- Holes at 0 and 3, VA at 5: f(x) = (x(x−3)(x+2)) / (x(x−3)(x−5)) → simplified = (x+2)/(x−5).
- Hole at 3, VAs at 0 and 5: f(x) = (x−3)(x+2) / (x(x−5)(x−3)) → simplified = (x+2)/(x(x−5)).

If you paste the function, I’ll factor it and tell you which statement is correct.