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Chapter 9: Problem 47

Graph each pair of functions using one set of axes. $$ f(x)=3^{x}, f^{-1}(x)=\log _{3} x $$

Short Answer

Expert verified
Graph \( f(x) = 3^x \) and \( f^{-1}(x) = \log_{3} x \) on the same axes, showing their reflection symmetry about the line \( y = x \).

Step by step solution

01

Understand the Functions

Identify the functions to be graphed: \( f(x) = 3^x \) and its inverse function \( f^{-1}(x) = \log_{3} x \).
02

Create a Table of Values for \( f(x) \)

Choose a range of \( x \) values and compute the corresponding \( f(x) \) values. For example: \( x = -2, -1, 0, 1, 2 \). Then, calculate \( f(-2) = 3^{-2} = \frac{1}{9} \), \( f(-1) = 3^{-1} = \frac{1}{3} \), \( f(0) = 3^0 = 1 \), \( f(1) = 3^1 = 3 \), \( f(2) = 3^2 = 9 \).
03

Create a Table of Values for \( f^{-1}(x) \)

Choose a range of \( x \) values for \( f^{-1}(x) \) and compute the corresponding values. Using the function \( f^{-1}(x) = \log_{3} x \), let \( x = \frac{1}{9}, \frac{1}{3}, 1, 3, 9 \). Then, calculate \( f^{-1}(\frac{1}{9}) = \log_{3} (\frac{1}{9}) = -2 \), \( f^{-1}(\frac{1}{3}) = \log_{3} (\frac{1}{3}) = -1 \), \( f^{-1}(1) = \log_{3} (1) = 0 \), \( f^{-1}(3) = \log_{3} (3) = 1 \), \( f^{-1}(9) = \log_{3} (9) = 2 \).
04

Plot the Points on the Graph

Plot the points from the tables created in steps 2 and 3 on the same set of axes. For example, for \( f(x) \), plot points \((-2, \frac{1}{9}), (-1, \frac{1}{3}), (0, 1), (1, 3), (2, 9) \). For \( f^{-1}(x) \), plot points \((\frac{1}{9}, -2), (\frac{1}{3}, -1), (1, 0), (3, 1), (9, 2) \).
05

Draw the Curves

Connect the points smoothly to draw the curves for both \( f(x) \) and \( f^{-1}(x) \). The graph of \( f(x) = 3^x \) will be an increasing exponential curve, and the graph of \( f^{-1}(x) = \log_{3} x \) will be an increasing logarithmic curve.
06

Verify Reflection Property

Check that the two graphs are symmetrical about the line \( y = x \). This verifies that \( f(x) \) and \( f^{-1}(x) \) are indeed inverse functions.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Exponential Functions
Exponential functions are mathematical expressions in which a constant base is raised to a variable exponent. In this case, our function's base is 3: \( f(x) = 3^x \). These functions are known for their rapid growth.
They have several key characteristics:
  • They always pass through the point (0,1) because any number raised to the power of 0 is 1.
  • They are always positive for any value of x.
  • As x increases, the function grows rapidly and moves away from the x-axis.
  • As x decreases, the function approaches the x-axis but never touches it, creating a horizontal asymptote at y = 0.
These properties together shape the exponential curve, giving it a consistently rising appearance.
Inverse Functions
Inverse functions reverse the effect of the original function. If you apply a function to a number and then apply its inverse function, you should return to the original number.
In this exercise, the inverse of the exponential function \( f(x) = 3^x \) is the logarithmic function \( f^{-1}(x) = \log_3 x \).
Inverse functions have important properties:
  • The graph of an inverse function is a reflection of the original function's graph across the line y = x.
  • They interchange the roles of x and y. For example, if \((a, b)\) is on the graph of f(x), then \((b, a)\) is on the graph of \( f^{-1}(x) \).
  • Applying an inverse function to the value given by the original function will yield the input value of the original function (i.e., \( f(f^{-1}(x)) = x \)).
Logarithmic Functions
Logarithmic functions are the inverses of exponential functions. For \( f(x) = 3^x \), the inverse function is \( f^{-1}(x) = \log_3 x \). This helps us find the exponent needed to achieve a certain base.
Here are the properties of logarithmic functions:
  • They pass through the point (1,0) because the log of 1 to any base is 0.
  • They are undefined for x 鈮� 0, meaning the function only takes positive values for x.
  • As x approaches zero from the right, the function goes to negative infinity.
  • As x increases, the function also increases but at a much slower rate, creating a vertical asymptote.
These characteristics create a curve that rises slowly and crosses the y-axis at 1.
Plotting Points
Plotting points involves calculating function values at specific x-values and then marking these points on a graph.
In this exercise, we calculated values for \( f(x) \) and \( f^{-1}(x) \) and plotted them:
  • For \( f(x) = 3^x \), we chose x-values like -2, -1, 0, 1, and 2 to get points: \( (-2, \frac{1}{9}), (-1, \frac{1}{3}), (0, 1), (1, 3), (2, 9) \).
  • For \( f^{-1}(x) \), we chose x-values like \( \frac{1}{9} \), \( \frac{1}{3} \), 1, 3, and 9 to get points: \( (\frac{1}{9}, -2), (\frac{1}{3}, -1), (1, 0), (3, 1), (9, 2) \).
Once points are plotted, you connect them smoothly, translating them into curves representing the functions.
For exponential functions, the curve rises fast, whereas for logarithmic functions, it rises slowly.
This visual representation makes it easier to understand how the functions behave and how they are related.

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