Question

Function 1:



Function 2:



Which statements comparing the graphs of function 1 and function 2 are true? Check all that apply.

Function 1 has a hole, while function 2 does not.
Function 1 has an oblique asymptote, while function 2 does not.
Function 2 has a hole, while function 1 only has asymptotes.
Function 2 only has asymptotes, while function 1 has a hole.
Both functions have the same domain restriction.
Both functions have the same vertical asymptote.

Answer 1

I don't see the two functions — the problem statement just shows blank "Function 1:" and "Function 2:". I can't decide which statements are true without the actual formulas or graphs.

If you paste the two function expressions (or images of the graphs), I can mark which statements are true. Meanwhile, here is a quick checklist you can use to test each statement yourself:

- Hole vs vertical asymptote:
- Factor numerator and denominator. If a factor (x − a) cancels (appears in both numerator and denominator) the graph has a removable discontinuity (a hole) at x = a.
- If a factor (x − a) remains in the denominator after cancellation, there is a vertical asymptote at x = a.
- Note: even if a factor cancels, x = a is still excluded from the domain of the original rational expression (it’s a hole, not allowed).

- Oblique (slant) asymptote:
- Perform polynomial long division. If deg(numerator) = deg(denominator) + 1, the quotient is linear and is the oblique asymptote. If deg(numerator) ≤ deg(denominator) there is no oblique asymptote; if deg(numerator) ≥ deg(denominator) + 2 the end behavior is polynomial of higher degree (no simple slant).

- “Only has asymptotes” vs “has a hole”:
- “Only has asymptotes” would mean there are no removable (canceled) factors — all denominator zeros produce vertical asymptotes (or possibly none if denominator never zero) and any nonvertical asymptote is horizontal/oblique as determined above.
- “Has a hole” means at least one factor cancels producing a removable discontinuity.

- Same domain restriction:
- Compare the set of x-values that make each denominator zero. Even canceled factors exclude that x from the function’s domain, so domain restriction is determined by all zeros of the original denominator for each function. If those zero sets are identical, the domain restrictions are the same.

- Same vertical asymptote:
- Compare the uncancelled denominator zeros of each function. If they share the same uncancelled zeros, they have the same vertical asymptotes.

If you paste the two functions, I’ll run through these checks and mark which of the given statements are true.