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Chapter 3: Problem 92

Graph \(f(x)=2^{x}\) and its inverse function in the same rectangular coordinate system.

Short Answer

Expert verified
The graph of \(f(x)=2^x\) is an increasing curve while the graph of its inverse function \(f^{-1}(x) = \log_2{x}\) is a decreasing curve. The two graphs are mirror images of each other about the line \(y = x\), illustrating that they are indeed inverse functions.

Step by step solution

01

Sketch the function \(f(x)=2^x\)

Start by creating a table of values for \(f(x) = 2^x\). For example, you can choose an array of integer values for \(x\) ranging from -2 to 2. Evaluate \(f(x)\) for each of the chosen \(x\) values. You'll find that as \(x\) increases, \(f(x)\) also increases since 2 raised to any power is always positive. Now, plot the function on a graph using the rectangle coordinate system.
02

Find the inverse function

The next step is determining the inverse of the function. For a function \(f(x) = 2^x\), its inverse function \(f^{-1}(x)\) is given by \(f^{-1}(x) = \log_2{x}\) because logarithms are the inverse operations of exponentials. Here, \(\log_2{x}\) is the power to which 2 must be raised to obtain the number \(x\).
03

Sketch the inverse function

Similar to the first step, create a table of values for the inverse function \(f^{-1}(x) = \log_2{x}\) by choosing \(x\) values and evaluating \(f^{-1}(x)\) for each. For this function, you might want to choose integer power of 2's (such as 0.25, 0.5, 1, 2, 4) for \(x\) to get neat integers for \(f^{-1}(x)\). Now, plot the values on the same rectangular coordinate system of the original function.
04

Check that they're inverse functions

Check that the orange graph (the inverse function, \(\log_2{x}\)) reflects the blue graph (the original function, \(f(x) = 2^x\)) over the line \(y = x\). The curves should mirror each other around the line \(y = x\). This is an indication that indeed \(\log_2{x}\) is the inverse of \(2^x\).

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