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Chapter 3: Problem 90

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.

Short Answer

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

Step by step solution

01

Understand the function \(f(x)=\left(\frac{1}{3}\right)^{x}\)

This function describes an exponential decay because the base \(\frac{1}{3}\) is less than 1. As the value of \(x\) increases, the value of \(f(x)\) decreases. The graph of this function descends from left to right.
02

Understand the function \(g(x)=3^{-x}\)

This function also describes an exponential decay. By using the law of exponents that states that \(a^{-x} = \frac{1}{a^x}\), we can rewrite the function as \(g(x)=\left(\frac{1}{3}\right)^{x}\), which is identical to the function \(f(x)\). Thus, its graph should be the same as that of \(f(x)\) and also descends from left to right.
03

Compare the two functions

By examining the formulas and the characteristics of the two functions, it can be concluded that they are indeed the same. Hence, the functions \(f(x)=\left(\frac{1}{3}\right)^{x}\) and \(g(x)=3^{-x}\) do have the same graph.

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