what are the domain and range of $f(x)=2(3^{x})$?\n domain: $(-\\infty,\\infty)$; range: $(0,\\infty)$\n…

what are the domain and range of $f(x)=2(3^{x})$?\n domain: $(-\\infty,\\infty)$; range: $(0,\\infty)$\n domain: $(-\\infty,\\infty)$; range: $(2,\\infty)$\n domain: $(0,\\infty)$; range: $(-\\infty,\\infty)$\n domain: $(2,\\infty)$; range: $(-\\infty,\\infty)$

what are the domain and range of $f(x)=2(3^{x})$?\n domain: $(-\\infty,\\infty)$; range: $(0,\\infty)$\n domain: $(-\\infty,\\infty)$; range: $(2,\\infty)$\n domain: $(0,\\infty)$; range: $(-\\infty,\\infty)$\n domain: $(2,\\infty)$; range: $(-\\infty,\\infty)$

Answer

Explanation:

Step1: Determine the domain

The exponential function $y = a^x$ ($a>0,a\neq1$) is defined for all real - valued $x$. Here, the function is $f(x)=2(3^x)$. Since $3^x$ is defined for all real $x$, the domain of $f(x)$ is all real numbers, which in interval notation is $(-\infty,\infty)$.

Step2: Determine the range

We know that for the exponential function $y = 3^x$, the range of $y = 3^x$ is $(0,\infty)$ because $3^x>0$ for all $x\in R$. Now, $f(x)=2(3^x)$. Since $3^x>0$, then $2(3^x)>0$. So the range of $f(x)$ is $(0,\infty)$.

Answer:

domain: $(-\infty,\infty)$; range: $(0,\infty)$