Question

Which statements are true of the graph of h(x) = RootIndex 3 StartRoot x minus 4 EndRoot? Check all that apply.

The domain of h(x) is the set of all real numbers.
The range of h(x) is the set of all real numbers.
For all points (x, h(x)), h(x) exists if and only if x – 4 0.
The graph of h(x) is a translation of f(x) down 4 units.
The graph of h(x) intercepts the x-axis at (4, 0).

Answer 1

True:
- The domain of h(x) is the set of all real numbers. (An odd root is defined for every real radicand.)
- The range of h(x) is the set of all real numbers. (∛ maps R onto R.)
- The graph of h(x) intercepts the x-axis at (4, 0). (Solve ∛(x−4)=0 ⇒ x=4.)

False:
- "h(x) exists iff x−4 ≥ 0." This is false because the cube root exists for all real x, not just x−4 ≥ 0.
- "The graph is a translation of f(x) down 4 units." This is false; h(x)=∛(x−4) is f(x)=∛x shifted right 4 units, not down.