sketch the graph of the function. state the domain and the range, and determine whether the function is…

Explanation:

Step1: Analyze the domain

For the exponential function (y = a^{x}) ((a>0,a\neq1)), the domain is all real numbers. So for (f(x)=\left(\frac{1}{11}\right)^{x}), the domain is ((-\infty,\infty)) since we can substitute any real - valued (x) into the function.

Step2: Analyze the range

Since (a=\frac{1}{11}>0), and for any real (x), (y = a^{x}>0). Also, as (x\to-\infty), (y=\left(\frac{1}{11}\right)^{x}=11^{-x}\to\infty) and as (x\to\infty), (y = \left(\frac{1}{11}\right)^{x}\to0). So the range is ((0,\infty))

Step3: Determine if the function is increasing or decreasing

We use the property of exponential functions (y = a^{x}). If (0 < a<1), the function (y=a^{x}) is decreasing. Here (a=\frac{1}{11}), and (0<\frac{1}{11}<1). Let (x_1<x_2). Then (f(x_1)-f(x_2)=\left(\frac{1}{11}\right)^{x_1}-\left(\frac{1}{11}\right)^{x_2}=\left(\frac{1}{11}\right)^{x_2}\left(11^{x_2 - x_1}-1\right)). Since (x_2-x_1>0), (11^{x_2 - x_1}>1) and (\left(\frac{1}{11}\right)^{x_2}>0), so (f(x_1)>f(x_2)) when (x_1 < x_2).

Answer:

• Domain: ((-\infty,\infty))
• Range: ((0,\infty))
• The function (f(x)=\left(\frac{1}{11}\right)^{x}) is decreasing.