Question

Consider the table representing a rational function.
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Which statement describes the graph of the function?
The function has holes when x = 0 and x = 4.
The function has vertical asymptotes when x = 0 and x = 4.
The function has a vertical asymptote when x = 0 and a hole when x = 4.
The function has a hole when x = 0 and a vertical asymptote when x = 4.

Answer 1

I can’t see the images you attached. To decide which statement is correct I need the table values (or a description of the y–values) for x near 0 and near 4.

How to tell from the table:
- If as x approaches a value c the y-values grow without bound (→ ±∞) or one side goes to +∞ and the other to −∞, then x = c is a vertical asymptote.
- If as x → c the y-values approach a finite number L, but the function value at c is missing or different, then x = c is a hole (a removable discontinuity).
- If as x → c the y-values approach a finite number L and f(c) = L, the function is continuous there.

If you paste the table rows (the x values and corresponding y values) or describe what the y-values do as x approaches 0 and 4, I’ll pick the correct statement. If you prefer, you can tell me just the values immediately to the left and right of 0 and 4 and whether f(0) and f(4) are defined.