Question

The function f(x)\:=\:x^{\frac{1}{3}} is transformed to get function h. h(x)\:=\:(2x)^{\frac{1}{3}}\: \:5 which statements are true about function h?

A. h(x) has a vertical shift downwards by 5 units.

B. The domain of h(x) is the same as the domain of f(x).

C. The graph of h(x) is stretched horizontally by a factor of 2.

D. The range of h(x) is the same as the range of f(x).

Answer 1

The domain of h(x) is the same as f(x), and the range remains unchanged. There is no horizontal stretch, but rather a horizontal compression. Additionally, the +5 indicates a vertical shift upwards, not downwards.

Step-by-step explanation:

When analyzing the function h(x) = (2x)^{\frac{1}{3}} + 5 and comparing it to the function f(x) = x^{\frac{1}{3}}, we can observe the following:

• Statement A suggests that there is a vertical shift downwards by 5 units, which is incorrect. The +5 actually indicates a vertical shift upwards by 5 units.
• Statement B states that the domain of h(x) is the same as the domain of f(x). This is true; the domain for both functions includes all real numbers since cube roots can be taken of negative, zero, and positive numbers.
• Statement C mentions that the graph of h(x) is stretched horizontally by a factor of 2. This is incorrect; changing the input to 2x actually compresses the graph horizontally by a factor of 1/2.
• Statement D suggests that the range of h(x) is the same as the range of f(x). This is true; the range is still all real numbers because the vertical shift does not affect the range of cubic root functions.