Problem 9 - Precalculus with Limits

Question

In Exercises 7-14, find the inverse function of \(f\) informally. Verify that \(f(f^{-1}(x)) = x\) and \(f^{-1}(f(x)) = x\). \(f(x) = x + 9\)

Step-by-Step Solution

Answer

The inverse function of \(f(x) = x + 9\) is \(f^{-1}(x) = x - 9\), and it can be confirmed that it satisfies the properties \(f(f^{-1}(x)) = x\) and \(f^{-1}(f(x)) = x\).

Step 1: Find the Inverse Function

To find the inverse of a function, swap the roles of \(x\) and \(y\). That is, replace every \(x\) in the function with a \(y\) and solve for \(y\). So, the function \(f(x) = x + 9\) becomes \(y = x + 9\). Swap \(x\) and \(y\) to get \(x = y + 9\). Now solve for \(y\) to get \(y = x - 9\). Thus, the inverse function, \(f^{-1}(x)\), is \(x - 9\).

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

Substitute \(f^{-1}(x)\) into \(f(x)\). That is, calculate \(f(f^{-1}(x))\). In this case, we have \(f(f^{-1}(x)) = (f^{-1}(x)) + 9 = (x - 9) + 9 = x\). Hence, \(f(f^{-1}(x)) = x\), which verifies the first property of inverse functions.

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

Substitute \(f(x)\) into \(f^{-1}(x)\). That is, calculate \(f^{-1}(f(x))\). In this case, we have \(f^{-1}(f(x)) = f(x) - 9 = (x + 9) - 9 = x\). Hence, \(f^{-1}(f(x)) = x\), which verifies the second property of inverse functions.

Key Concepts

Verifying Inverse Functions

When you find the inverse of a function, you are essentially reversing its operation. To make sure that your inverse function is correct, you'll need to verify it. This means checking if the inverse function truly 'undoes' the original function. There are two important checks:

• First, plug the inverse function into the original function. You should get the input value back, symbolized as: \(f(f^{-1}(x)) = x\).
• Second, do the reverse by placing the original function inside the inverse function. It should also return the input: \(f^{-1}(f(x)) = x\).

Understanding that these two criteria must hold helps in confirming that the functions are indeed inverses of each other. Think of it as a puzzle where the pieces must fit perfectly in both scenarios. If both conditions check out, you've successfully verified that your inverse function works correctly.

Function Composition

Function composition is when one function is applied to another. Imagine you have two functions, say \(f(x)\) and \(g(x)\). If you apply \(g(x)\) and then apply \(f(x)\) to the result of \(g(x)\), that's composition. It's represented as \((f \circ g)(x)\) or simply \(f(g(x))\). When verifying inverses, composition helps check if they cancel each other.In the scenario of verifying inverses,

• The composition \(f(f^{-1}(x)) = x\) involves replacing every \(x\) in \(f\) with \(f^{-1}(x)\). This should give you back just \(x\).
• Conversely, \(f^{-1}(f(x)) = x\) replaces every \(x\) in \(f^{-1}\) with \(f(x)\), also resulting in \(x\).

Learning to compose functions reinforces how operations can be systematically reversed and understood through these manipulations, making it a powerful tool in algebra.

Algebraic Manipulation

Algebraic manipulation is the process of rearranging and simplifying expressions to solve equations. It's a core skill for solving problems like finding inverse functions. Let’s take a look at how it works in our exercise.To find the inverse of a function like \(f(x) = x + 9\), algebraic manipulation steps include:

• Replace \(f(x)\) with \(y\), giving \(y = x + 9\).
• Swap the variable positions to get \(x = y + 9\).
• Finally, solve for \(y\) by isolating it, transforming the equation into \(y = x - 9\).

This way of manipulating equations through operations like adding, subtracting, multiplying, or dividing helps reveal an inverse function step by step. Mastering these techniques enables you to handle and solve various mathematical challenges with ease.