what are the domain and range of $f(x)=\\left(\\frac{1}{5}\\right)^x$?\nthe domain is all real numbers. the…

Explanation:

Step1: Recall domain of exponential - function

For an exponential function of the form $y = a^{x}$ ($a>0,a\neq1$), $x$ can take any real - value. Here $a=\frac{1}{5}$ and the function is $f(x)=(\frac{1}{5})^{x}$. So the domain is all real numbers, i.e., $x\in(-\infty,\infty)$.

Step2: Analyze the range of the exponential function

Since $a = \frac{1}{5}\in(0,1)$, as $x\rightarrow-\infty$, $(\frac{1}{5})^{x}\rightarrow\infty$, and as $x\rightarrow\infty$, $(\frac{1}{5})^{x}\rightarrow0$. Also, for any real number $x$, $(\frac{1}{5})^{x}>0$. So the range of the function $y = (\frac{1}{5})^{x}$ is all real numbers greater than 0, i.e., $y\in(0,\infty)$.

Answer:

The domain is all real numbers. The range is all real numbers greater than zero.