Question - Logarithmic Function Transformation Analysis
Solution:
The given logarithmic function f(x) = log(x) is transformed to g(x) = log(x + 1) + 3. To determine which of the statements is true, we need to analyze the transformation that g(x) represents compared to f(x).- For statement A: f(x) is translated 1 unit upward. This is not true because the function g(x) has a "+3" outside of the logarithm, which means the entire function is translated 3 units upward, not 1 unit.- For statement B: f(x) is translated 3 units downward. This is incorrect as the transformation involves a "+3", indicating an upward translation, not a downward one.- For statement C: The vertical asymptote shifts 1 unit to the left. This is true. The vertical asymptote for the basic logarithmic function f(x) = log(x) is at x=0. Since g(x) is log(x + 1), this transformation shifts the graph horizontally 1 unit to the left, meaning the new vertical asymptote is now at x=-1.- For statement D: The vertical asymptote shifts 3 units to the right. This is not true. The transformation inside the logarithmic function (x + 1) does not shift the graph to the right; it shifts it to the left. Hence, the correct statement is C: The vertical asymptote shifts 1 unit to the left.
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