Question

f(x)= root(-6)(x+21)

Answer 1

I want to be sure I understand your notation. "root(-6)(x+21)" can be read in two common ways:

1) f(x) = (x+21)^{1/(-6)} = (x+21)^{-1/6} = 1/(x+21)^{1/6} (the −6th root, i.e. reciprocal of the 6th root), or
2) f(x) = −(x+21)^{1/6} (a minus sign in front of the 6th root).

Which did you mean? Below are brief facts for each.

If f(x) = (x+21)^{-1/6}:
- Domain: x > −21 (radicand ≥0 but radicand = 0 gives division by zero).
- Range: (0, ∞).
- Behavior: f(x) → +∞ as x → −21^+; f(x) → 0^+ as x → +∞.
- Monotonicity: decreasing for x > −21 (f'(x) = −(1/6)(x+21)^{-7/6} < 0).
- Vertical asymptote at x = −21; horizontal asymptote y = 0.

If f(x) = −(x+21)^{1/6}:
- Domain: x ≥ −21.
- Range: (−∞, 0].
- Behavior: f(−21) = 0; f(x) → −∞ as x → +∞.
- Monotonicity: decreasing for x ≥ −21 (f'(x) = −(1/6)(x+21)^{-5/6} < 0 except at −21).
- No vertical asymptote; no finite horizontal asymptote.

Tell me which form you intended (or paste the expression in another form), and I can plot it, find an inverse if it exists, or do whatever analysis you need.