Question

Graph each function and its inverse on the same set of axes. $$ f(x)=4^{x} ; f^{-1}(x)=\log _{4} x $$

Short answer

Plot \( f(x) = 4^x \) and \( f^{-1}(x) = \log_{4} x \) with points and reflect \( y = x \).

Step-by-step solution

Understand the Functions

The original function is given by \( f(x) = 4^x \) and its inverse is \( f^{-1}(x) = \log_{4} x \). The goal is to plot both the function and its inverse on the same set of axes.

Choose a Range for x

Select a reasonable range for \( x \) to plot the functions. Common choices are from \( -2 \) to \( 2 \) for exponential functions, and a similar range for logarithmic functions.

Create a Table of Values for f(x)

Calculate \( y \) values for \( f(x) = 4^x \) at selected \( x \) values. For example:\( x = -2, -1, 0, 1, 2 \)\( y = 4^{-2}, 4^{-1}, 4^0, 4^1, 4^2 \)This results in the pairs: (-2, 1/16), (-1, 1/4), (0, 1), (1, 4), (2, 16).

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

Calculate \( y \) values for \( f^{-1}(x) = \log_{4}(x) \) at selected \( x \) values. For example:\( x = 1/16, 1/4, 1, 4, 16 \)\( y = \log_{4}(1/16), \log_{4}(1/4), \log_{4}(1), \log_{4}(4), \log_{4}(16) \)This results in the pairs: (1/16, -2), (1/4, -1), (1, 0), (4, 1), (16, 2).

Plot the Function f(x) and Its Inverse f^{-1}(x)

Using the tables of values, plot the points for \( f(x) = 4^x \) and connect them with a smooth curve. Do the same for \( f^{-1}(x) = \log_{4} x \). Remember that the graph of \( f^{-1}(x) \) is a reflection over the line \( y = x \).

Draw the Line y = x

Draw the line \( y = x \) on the graph. This line will help visualize the reflection property between a function and its inverse.

Key concepts

Exponential Functions

An exponential function is a mathematical function of the form \( f(x) = a^x \), where \( a \) is a positive constant different from 1. In the given problem, our function is \( f(x) = 4^x \). Exponential functions grow quickly when \( x \) is positive and decrease rapidly to 0 as \( x \) becomes negative. The base 4 means that the function value quadruples for each unit increase in \( x \). Exponential functions have a horizontal asymptote at \( y = 0 \), meaning as \( x \) approaches negative infinity, \( f(x) \) approaches 0 but never actually touches the x-axis.

Logarithmic Functions

A logarithmic function is the inverse of an exponential function and is represented as \( f^{-1}(x) = \log_b(x) \), where \( b \) is the base of the logarithm. In our example, the inverse function is \( f^{-1}(x) = \log_4(x) \). Logarithmic functions grow slowly and are the 'mirror image' of their exponential counterparts over the line \( y = x \). They are defined only for positive values of \( x \). Logarithmic functions have a vertical asymptote at \( x = 0 \), meaning as \( x \) approaches 0 from the right, \( f^{-1}(x) \) approaches negative infinity.

Inverse Functions

Inverse functions effectively 'reverse' the effect of the original function. If \( y = f(x) \), then \( x = f^{-1}(y) \). For example, in our problem, if \( f(x) = 4^x \), then \( f^{-1}(x) = \log_4(x) \). The graph of an inverse function is obtained by reflecting the graph of the original function over the line \( y = x \). This is because for every point \( (a, b ) \) on the graph of \( f(x) \), there is a corresponding point \( (b, a) \) on the graph of \( f^{-1}(x) \). This reflective relationship helps visualize why functions and their inverses appear as mirror images with respect to \( y = x \).

Graphing Functions

Graphing is a crucial skill for understanding the behavior of functions. To graph \( f(x) = 4^x \) and \( f^{-1}(x) = \log_4(x) \):
1. Choose a suitable range for \( x \). For example, from \( -2 \) to \( 2 \) for \( f(x) = 4^x \) and from \( 1/16 \) to \( 16 \) for \( f^{-1}(x) = \log_4(x) \).
2. Create a table of values for each function. Evaluate their output for specific \( x \) values.
3. Plot the points on a coordinate plane.
4. Connect the points smoothly to outline the curves.
5. Draw the line \( y = x \), which will help visualize the reflection property between the function and its inverse.

Reflection Property

The reflection property states that the graph of a function and its inverse are mirror images over the line \( y = x \). This means if you take a point on the graph of the function \( f(x) \) and reflect it over the line \( y = x \), you will get a corresponding point on the graph of its inverse function, \( f^{-1}(x) \). Understanding this property is essential because:

• It helps you graph inverses accurately.
• It provides a visual confirmation that two functions are indeed inverses of each other.
• It demonstrates how values interchange between functions and their inverses, reinforcing the concept of 'undoing' an operation.