Question

Consider the table representing a rational function.
mc029-1.jpg mc029-2.jpg mc029-3.jpg mc029-4.jpg mc029-5.jpg mc029-6.jpg mc029-7.jpg mc029-8.jpg mc029-9.jpg mc029-10.jpg mc029-11.jpg mc029-12.jpg mc029-13.jpg mc029-14.jpg mc029-15.jpg mc029-16.jpg mc029-17.jpg
mc029-18.jpg mc029-19.jpg mc029-20.jpg mc029-21.jpg undefined mc029-22.jpg mc029-23.jpg mc029-24.jpg mc029-25.jpg undefined mc029-26.jpg mc029-27.jpg mc029-28.jpg undefined mc029-29.jpg mc029-30.jpg mc029-31.jpg

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 can’t see the images you attached. Could you either paste the table values here or type the rows/columns (x and f(x) values) shown near x = 0, 3, and 5?

If you want to decide it yourself, use this rule:
- If as x approaches a value a from both sides f(x) approaches a finite number L, but f is undefined at x = a (or f(a) ≠ L), then x = a is a hole.
- If as x approaches a from one or both sides f(x) grows without bound (→ ±∞) or the left- and right-hand values go to opposite infinities, then x = a is a vertical asymptote.

So look at the table entries for x values just left and right of 0, 3, and 5:
- If the y-values approach a finite number on both sides → hole.
- If the y-values blow up (very large magnitude or sign changes to ±∞) → vertical asymptote.

Share the nearby table values and I’ll pick which of the four statements is correct.