Identifying Transformed Exponential Function
The question asks which function g(x) represents the exponential function f(x) = 5^x after a vertical stretch by a factor of 2 and a reflection across the x-axis.
Let's break down the transformations:
1. Vertical stretch by a factor of 2: This means we multiply the original function f(x) by 2. So the function becomes 2 * 5^x.
2. Reflection across the x-axis: To reflect a function across the x-axis, we take the negative of the function. So the function now becomes -2 * 5^x.
Now, let's look at the answer choices and see which one matches our transformed function:
A. g(x) = 2 * 5^(-x) - This represents a horizontal reflection across the y-axis instead of a vertical reflection and stretch. So, this is incorrect.
B. g(x) = -2 * 5^(-x) - This represents a reflection across the x-axis and y-axis. So, this is incorrect.
C. g(x) = 2 * 5^x - This represents a vertical stretch but no reflection across the x-axis. So, this is incorrect.
D. g(x) = -2 * 5^x - This matches our description of a vertical stretch by a factor of 2 and a reflection across the x-axis.
Therefore, the correct answer is D: g(x) = -2 * 5^x.