Analyze the effect on the domain and range of $$ f(x)=\ln (x) $$ when $$ f(x) $$ is replaced by $$ a f(x) $$ where $$ a

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Analyze the effect on the domain and range of $$ f(x)=\ln (x) $$ when $$ f(x) $$ is replaced by $$ a f(x) $$ where $$ a<0 $$. Option \#1: Neither the domain nor the range change. Option \#2: The domain changes, but the range does not. Option \#3: The range changes, but the domain does not. Option \#4: Both the domain and the range change. (1 point) Option \# describes the effect on the domain and range of $$ f(x)=\ln (x) $$ when $$ f(x) $$ is replaced

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The function f(x) = ln(x) has a domain of (0, ∞) and a range of (–∞, ∞).

When you multiply by a constant a where a < 0, the operation becomes g(x) = a·ln(x). Multiplying by a constant does not affect the x-values for which ln(x) is defined. Therefore, the domain remains (0, ∞).

Since ln(x) can take any real value, multiplying by any nonzero constant (even a negative one) will still produce all real numbers (although the output will be "stretched" and reflected). Thus, the range remains (–∞, ∞).

So, the correct option is Option #1: Neither the domain nor the range change.
Neither the domain nor the range of $$ f(x) = \ln(x) $$ changes when multiplied by a negative constant $$ a $$.

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