Problem 10 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the inverse function of \(f\) informally. Verify that \(f\left(f^{-1}(x)\right)=x\) and \(f^{-1}(f(x))=x.\) $$f(x)=x+3$$

Step-by-Step Solution

Answer

The inverse function of \(f(x) = x + 3\) is \(f^{-1}(x) = x - 3\). Both properties \(f\left(f^{-1}(x)\right) = x\) and \(f^{-1}\left(f(x)\right) = x\) hold true.

Step 1: Find the Inverse Function

To find the inverse of a function, replace \(f(x)\) with \(y\), and then swap \(x\) and \(y\). Thus, if \(f(x) = x + 3\) is replaced with \(y = x + 3\), swapping \(x\) and \(y\) gives \(x = y + 3\). Solving this for \(y\) to express \(y = f^{-1}(x)\) in terms of \(x\) gives \(y = x - 3\). Therefore, \(f^{-1}(x) = x - 3\).

Step 2: Verify \(f(f^{-1}(x)) = x\)

Substitute \(f^{-1}(x)\) into the function \(f\). Thus, we replace \(x\) in the function \(f(x) = x + 3\) with \(f^{-1}(x) = x - 3\) to get \(f\left(f^{-1}(x)\right) = (x - 3) + 3\). Simplifying this gives \(x\). So, the property \(f\left(f^{-1}(x)\right) = x\) holds true.

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

Substitute \(f(x)\) into the function \(f^{-1}\). Replace \(x\) in the function \(f^{-1}(x) = x - 3\) with \(f(x) = x + 3\) to get \(f^{-1}\left(f(x)\right) = (x + 3) - 3\). Simplifying this gives \(x\). So, the property \(f^{-1}\left(f(x)\right) = x\) also holds true.

Key Concepts

Function Composition

Function composition is a crucial concept in mathematics that involves creating a new function by combining two existing functions. When you hear function composition, think of it as inserting one function into another.
In mathematical terms, if you have two functions, say \( f(x) \) and \( g(x) \), the composition of these functions is written as \( (f \circ g)(x) = f(g(x)) \). Essentially, you take the output of \( g(x) \) and use it as the input for \( f(x) \).

• For example, with \( f(x) = x + 3 \) and \( g(x) = x - 3 \), the composition \( f(g(x)) \) becomes \( f(x - 3) = (x - 3) + 3 = x \).
• Similarly, composing in the other order, \( g(f(x)) = g(x + 3) = (x + 3) - 3 = x \).

It turns out that, with this composition, the inverse relationships between functions are beautifully highlighted. If such compositions yield the original input variable \( x \), then the functions are inverses of each other.

Verification of Inverses

Verifying that two functions are inverses involves showing that both possible compositions of these functions return the identity function—that is, they return the original input value.
For instance, if \( f(x) \) has an inverse \( f^{-1}(x) \), confirming that the functions are inverses requires checking two conditions:

• First, if \( f(f^{-1}(x)) = x \), it means that applying \( f \) to \( f^{-1}(x) \) results in the initial \( x \). For the function \( f(x) = x + 3 \), let's check: \( f(f^{-1}(x)) = f(x - 3) = (x - 3) + 3 = x \).
• Second, if \( f^{-1}(f(x)) = x \), then applying \( f^{-1} \) to \( f(x) \) also leads back to the starting value \( x \). With \( f^{-1}(x) = x - 3 \), checking gives: \( f^{-1}(f(x)) = f^{-1}(x + 3) = (x + 3) - 3 = x \).

When both these conditions hold, \( f(x) \) and \( f^{-1}(x) \) act as perfect inverses, effectively undoing each other's operations.

Linear Functions

Linear functions are perhaps the simplest type of functions you will encounter in algebra. A linear function is one whose graph is a straight line and can be expressed in the form \( f(x) = ax + b \), where \( a \) and \( b \) are constants.
What makes linear functions interesting is their consistent rate of change, represented by the slope \( a \).

• In the given function \( f(x) = x + 3 \), the slope \( a \) is 1, meaning that for every unit increase in \( x \), \( f(x) \) increases by 1 as well.
• The constant term \( b \) is 3, which tells us where the line crosses the y-axis (the vertical line when \( x = 0 \)).

Ultimately, the simplicity of linear functions makes them an excellent tool for understanding more complex mathematical concepts, such as inverses and function compositions. They offer a perfect starting point for exploring how functions can represent real-world relationships.