Final answer:
Of the statements provided, only Statement B is true, indicating a vertical shrink by a factor of 1/2 for the function f(x)=1/2(3)√x compared to f(x)= 3√x. Statements A, C, and D are false due to incorrect assumptions about domain, range, and the nature of horizontal transformations.
Step-by-step explanation:
Looking at the statements given, we can deduce the truth of each:
- A. False. The function F(x)=1/2 (3)√(x-2) has a domain of x ≥ 2 since you cannot take a cube root of a negative number without involving complex numbers. In contrast, f(x)= 3√x has a domain of x ≥ 0 because the cube root can also be taken of negative numbers. Their ranges are similar but their domains are different.
- B. True. Multiplying a function by 1/2 will compress it vertically by a factor of 1/2. So, the graph of f(x)=1/2(3)√x will indeed look like the graph of f(x)= 3√x, with a vertical compression.
- C. False. Horizontal transformations are not achieved by simply multiplying the function by a constant. The actual horizontal stretch or shrink involves transforming the x-values within the function itself.
- D. False. As with statement B, multiplying by 1/2 does result in a vertical shrink. However, F(x)=1/2√x is not the same function as f(x)= 3√x and it does not represent just a vertical shrink of the graph of f(x)= 3√x due to the different coefficients of x within the cube root. The graph of f(x)=1/2√x is a vertical shrink of the basic cube root graph, not the graph of f(x)= 3√x.
In the context of two-dimensional (x-y) graphing, the coefficient of a function reflects vertical stretches or compressions, while horizontal transformations involve changes inside the function's argument.