Final answer:
The function g(x) can be found by analyzing the given transformations separately and understanding how they affect the original function f(x) = x^3. The correct option is B) g(x) = -5(x - 4)^3.
Step-by-step explanation:
The given functions are transformations of the function f(x) = x^3. We need to find a formula for the function g(x) that represents the given transformations in options A, B, C, and D:
A) g(x) = 5(x + 4)^3
B) g(x) = -5(x - 4)^3
C) g(x) = -5(x + 4)^3
D) g(x) = 5(x - 4)^3
To find g(x) for each option, we need to analyze the transformation separately. The number in front of the function represents a vertical stretch or compression, while the expression inside the function represents a horizontal translation.
A) In option A, the function is stretched vertically by a factor of 5. The expression (x + 4) inside the function represents a horizontal translation of 4 units to the left.
B) In option B, the function is stretched vertically by a factor of -5, which also reflects the function across the x-axis. The expression (x - 4) inside the function represents a horizontal translation of 4 units to the right.
C) In option C, the function is stretched vertically by a factor of -5 and reflected across the x-axis. The expression (x + 4) inside the function represents a horizontal translation of 4 units to the left.
D) In option D, the function is stretched vertically by a factor of 5. The expression (x - 4) inside the function represents a horizontal translation of 4 units to the right.
Therefore, based on the given transformations, the correct option is B) g(x) = -5(x - 4)^3.