Problem 8 - Precalculus with Limits

Question

In Exercises 7-14, find the inverse function of \(f\) informally. Verify that \(f(f^{-1}(x)) = x\) and \(f^{-1}(f(x)) = x\). \(f(x) = \frac{1}{3}x\)

Step-by-Step Solution

Answer

The inverse of the function \(f(x) = \frac{1}{3}x\) is \(f^{-1}(x) = 3x\). The properties \(f(f^{-1}(x)) = x\) and \(f^{-1}(f(x)) = x\) hold true for this function and its inverse.

Step 1: Identifying The Function

The function given is \(f(x) = \frac{1}{3}x\). The goal is to find the inverse of this function.

Step 2: Finding the Inverse of the Function

To find the inverse function, you interchange the x and y in the equation and solve for y. Doing this gives: \(x = \frac{1}{3}y\). Solving for y by multiplying both sides by 3, you obtain \(y = 3x\). Therefore, the inverse of \(f(x)\), denoted as \(f^{-1}(x)\), is \(f^{-1}(x) = 3x\).

Step 3: Verifying that \(f(f^{-1}(x)) = x\)

To do this, substitute \(f^{-1}(x)\) which is 3x into f(x). This gives: \(f(f^{-1}(x)) = \frac{1}{3}*(3x) = x\).

Step 4: Verifying that \(f^{-1}(f(x)) = x\)

Substitute \(f(x)\) which is \(\frac{1}{3}x\) into \(f^{-1}(x)\) to obtain: \(f^{-1}(f(x)) = 3*\frac{1}{3}x = x\).

Key Concepts

Function Composition

Function composition is an important concept in mathematics. It involves the combining of two functions to form a new function. When we have two functions, say \( g(x) \) and \( f(x) \), the function composition \( f(g(x)) \) means substituting the entire function \( g(x) \) wherever there is an \( x \) in the function \( f(x) \).
This process allows us to see how the output of one function becomes the input of another. For example, if \( f(x) = \frac{1}{3}x \) and \( g(x) = 3x \), the composition \( f(g(x)) \) would be \( \frac{1}{3}(3x) \).
By understanding function composition, you can see how functions interact with each other. This is crucial for verifying inverse functions, where we typically use this composition process.

Verification of Inverse

Verifying that two functions are inverses of each other means confirming that they "undo" each other. For two functions \( f \) and \( f^{-1} \) to be inverses, they must satisfy:\

• \( f(f^{-1}(x)) = x \)
• \( f^{-1}(f(x)) = x \)

When one function is applied after the other, the original input \( x \) should be returned.
In our example, we have \( f(x) = \frac{1}{3}x \) and its inverse \( f^{-1}(x) = 3x \). To verify, we first substitute the inverse into the function: \( f(f^{-1}(x)) = \frac{1}{3}(3x) = x \). This shows that applying \( f \) after \( f^{-1} \) returns \( x \).
Next, we substitute \( f(x) \) into its inverse: \( f^{-1}(f(x)) = 3(\frac{1}{3}x) = x \). Both processes confirm the inverse relationship. This verification is key in confirming that two functions are essentially reversing each other's effects.

Linear Functions

Linear functions are among the simplest types of functions in mathematics. A linear function has the general form \( f(x) = ax + b \), where \( a \) and \( b \) are constants. The graph of a linear function is always a straight line, which makes them easy to understand and manipulate.
In the given exercise, the function \( f(x) = \frac{1}{3}x \) is a linear function because it fits the standard form with \( a = \frac{1}{3} \) and \( b = 0 \). Linear functions are often used in initial algebra studies because their behavior is predictable and straightforward.
They are pivotal for finding inverse functions since their inverses are also linear. For \( f(x) = \frac{1}{3}x \), the inverse is \( f^{-1}(x) = 3x \), which showcases the simplicity and symmetry often characteristic of linear functions.