Question

Which is the graph of f(x) = RootIndex 3 StartRoot x EndRoot?

Answer 1

f(x) = ∛x (the cube root of x).

Key features
- Domain = (−∞, ∞), Range = (−∞, ∞).
- Passes through (−1, −1), (0, 0), (1, 1) (and e.g. (−8, −2), (8, 2)).
- Odd function (symmetric about the origin).
- Monotone increasing for all x.
- Vertical tangent at x = 0 (slope → ∞ there).
- Concave up for x < 0, concave down for x > 0, with an inflection at (0,0).
- No asymptotes; grows slowly as |x| → ∞.

Rough sketch (y on vertical, x on horizontal)

y
|
2 | .
| .
1 | .
| .
0 ---+----.----------- x
| .
-1 | .
|.
-2 .

The curve is the familiar S-shaped cube-root curve through the origin, increasing and odd.