Question

Which statements describe the function ? Check all that apply.

f(x) has a vertical asymptote at x = 8.
f(x) has a vertical asymptote at x = 12.
f(x) has a hole at x = –12.
f(x) has a horizontal asymptote at y = 0.
f(x) has an x-intercept at (–24, 0).
f(x) is increasing from (8, ∞).

Answer 1

I don't see the function or its graph — I need that to determine which statements are true. Please paste the formula for f(x) or upload the graph.

Meanwhile, here are quick rules you can use to check each statement (most are for rational functions f(x) = N(x)/D(x)):

- "Vertical asymptote at x = a": D(a) = 0 but N(a) ≠ 0 after cancelling any common factors. If a factor cancels (common zero), it's not a vertical asymptote — it may be a hole.

- "Hole at x = a": both N(a) = 0 and D(a) = 0 (a common factor), so the factor cancels and the simplified function is defined by a removable discontinuity at a.

- "Horizontal asymptote at y = 0": degree(N) < degree(D). If degrees equal, horizontal asymptote is y = leading coeff(N)/leading coeff(D); if degree(N) > degree(D) by ≥1, no horizontal asymptote (maybe an oblique asymptote).

- "x-intercept at (a,0)": f(a) = 0 (after cancelling holes). That means N(a) = 0 but D(a) ≠ 0.

- "f is increasing on (8, ∞)": compute f'(x) and check its sign for x > 8 (or inspect the graph).

Give the function (or graph) and I’ll evaluate each statement.