The function f(x) = (1/2)^x on R is | vedclass

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(A) Let $f(x) = (1/2)^x$ for all $x \in R$. To determine the nature of the function,we find its derivative: $f'(x) = \frac{d}{dx} \left( (1/2)^x \righ

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Strictly decreasing

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(A) Let $f(x) = (1/2)^x$ for all $x \in R$. To determine the nature of the function,we find its derivative: $f'(x) = \frac{d}{dx} \left( (1/2)^x \right) = (1/2)^x \ln(1/2)$. Since $\ln(1/2) = \ln(1) - \ln(2) = 0 - \ln(2) = -\ln(2)$,we have: $f'(x) = -(1/2)^x \ln(2)$. Since $(1/2)^x > 0$ for all $x \in R$ and $\ln(2) > 0$,it follows that $f'(x) Because the derivative $f'(x)$ is strictly less than $0$ for all $x$ in the domain,the function $f(x)$ is strictly decreasing on $R$.

The function $f(x) = (1/2)^x$ on $R$ is

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