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.
To solve this problem, we'll consider the transformations required step by step. The given function is f(x) = 5^x, and we are asked to apply the following transformations: 1. Vertical stretch by a factor of 2. 2. Reflection across the x-axis. 1. A vertical stretch by a factor of 2 will multiply the function by 2. This does not affect the exponent, so the new function after this transformation will be: h(x) = 2 * f(x) = 2 * 5^x 2. A reflection across the x-axis means we need to take the negative of the function. So the final function after this reflection will be: g(x) = -h(x) = -2 * 5^x This matches option B: g(x) = -2(5^x).
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