Question

Sketch the graphs of the function \(y = {3^x},\;y = {10^x},\;y = {\left( {\frac{1}{3}} \right)^x}\)and \(y = {\left( {\frac{1}{{10}}} \right)^x}\) on the same axes and interpret how these graphs are related.

Short answer

The graphs of the functions \(y = {3^x},\,y = {10^x}\) are monotonically increasing and the graphs of the functions \(y = {\left( {\frac{1}{3}} \right)^x}\) and \(y = {\left( {\frac{1}{{10}}} \right)^x}\) are monotonically decreasing.

Step-by-step solution

Given data

The functions are\(y = {3^x},\;y = {10^x},\;y = {\left( {\frac{1}{3}} \right)^x}\)and \(y = {\left( {\frac{1}{{10}}} \right)^x}\).

Concept of the domain of an exponential function

In general, the domain of the function is the set of all independent value that defines that function.

The domain of an exponential functions\(y = {a^x},\;a > 0\)and\(a \ne 1\)is the set of real values.

That is,\( - \infty < x < \infty \).

Sketch the graphs of the function \(y = {3^x},\;y = {10^x},\;y = {\left( {\frac{1}{3}} \right)^x}\)and\(y = {\left( {\frac{1}{{10}}} \right)^x}\)

The rough graphs of the functions \(y = {3^x},\;y = {10^x},\;y = {\left( {\frac{1}{3}} \right)^x}\) and \(y = {\left( {\frac{1}{{10}}} \right)^x}\) are in figure 1 as follows:

From figure 1, it is observed that the graphs of the functions are exponential function.

Also, identify that all the graphs are passing through the common point \((0,1)\) as any number raised to the power 0 is \(1\).

Therefore, the graphs of the functions \(y = {3^x},\;y = {10^x}\) are monotonically increasing and the graphs of the functions \(y = {\left( {\frac{1}{3}} \right)^x}\) and \(y = {\left( {\frac{1}{{10}}} \right)^x}\) are monotonically decreasing.