Problem 78 - Introductory and Intermediate Algebra for College Students 4th

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The functions \(f(x)=\left(\frac{1}{3}\right)^{x}\) and \(g(x)=3^{-x}\) have the same graph.

Step-by-Step Solution

Answer

The statement is true. The functions \(f(x)=\left(\frac{1}{3}\right)^{x}\) and \(g(x)=3^{-x}\) are the same and they would indeed have the same graph.

Step 1: Understand the given functions

The functions provided are \(f(x) = (1/3)^x\) and \(g(x) = 3^{-x}\). The statement claims that these are equivalent and would produce the same graph.

Step 2: Rewriting function g

The function \(g(x) = 3^{-x}\) can be rewritten by recalling a rule of exponents that says \(a^{-n} = 1/a^n\). Therefore, \(3^{-x}=1/3^x\). Now you can see that \(f(x) = g(x)\) and therefore they would indeed have the same graph.