Problem 55 - Intermediate Algebra

Question

Show that each pair of functions are inverses. $$ f(x)=2 x+9, f^{-1}(x)=\frac{x-9}{2} $$

Step-by-Step Solution

Answer

The functions \( f(x) = 2x + 9 \) and \( f^{-1}(x) = \frac{x-9}{2} \) are inverses.

Step 1: Understand the Definition of Inverse Functions

Two functions \( f \) and \( g \) are inverses of each other if and only if \( f(g(x)) = x \) and \( g(f(x)) = x \). This means applying one function and then applying the inverse function will return the original input value.

Step 2: Compose f and f^{-1}

To confirm that \( f(x) = 2x + 9 \) and \( f^{-1}(x) = \frac{x - 9}{2} \) are inverses, compute \( f(f^{-1}(x)) \). This means substitute \( f^{-1}(x) \) into \( f(x) \): \[ f(f^{-1}(x)) = f\left(\frac{x-9}{2}\right) = 2\left(\frac{x-9}{2}\right) + 9. \]

Step 3: Simplify the Composition f(f^{-1}(x))

Simplify the expression: \[ 2\left(\frac{x-9}{2}\right) + 9 = x - 9 + 9 = x. \]Thus, \( f(f^{-1}(x)) = x \). This confirms one part of the inverse relationship.

Step 4: Compose f^{-1} and f

Next, compute \( f^{-1}(f(x)) \) by substituting \( f(x) \) into \( f^{-1}(x) \): \[ f^{-1}(f(x)) = f^{-1}(2x + 9) = \frac{(2x + 9) - 9}{2}. \]

Step 5: Simplify the Composition f^{-1}(f(x))

Simplify the expression to verify it simplifies to \( x \): \[ \frac{(2x + 9) - 9}{2} = \frac{2x}{2} = x. \]So, \( f^{-1}(f(x)) = x \). This confirms the other part of the inverse relationship.

Step 6: Conclude that the Functions are Inverses

Since both compositions \( f(f^{-1}(x)) \) and \( f^{-1}(f(x)) \) simplify to \( x \), the functions \( f(x) = 2x + 9 \) and \( f^{-1}(x) = \frac{x-9}{2} \) are confirmed to be inverses of each other.

Key Concepts

Function Composition

Function composition is a crucial concept in understanding inverse functions. It involves the process of applying one function to the results of another function. In terms of notation, if you have two functions, say \( f(x) \) and \( g(x) \), their composition is represented as \( f(g(x)) \) or \( g(f(x)) \). This implies that you first apply function \( g \), then apply function \( f \) to the result of \( g \). When dealing with inverse functions, composition helps verify the relationship between the two functions.
To determine if two functions are inverses of each other, we use composition to check two conditions:
1. \( f(g(x)) = x \)
2. \( g(f(x)) = x \)
If both conditions are met, one function is the inverse of the other. As an example, for \( f(x) = 2x + 9 \) and its supposed inverse \( f^{-1}(x) = \frac{x - 9}{2} \), validating these conditions through composition proves they are indeed inverse functions.

Algebraic Functions

Algebraic functions are expressions that involve operations like addition, subtraction, multiplication, division, and taking roots. These functions are central to many areas of mathematics and often involve solving equations to find roots or simplify expressions. Understanding algebraic functions is key to manipulating and forming the inverses. In our example, we have \( f(x) = 2x + 9 \), a simple linear algebraic function.
To find the inverse of an algebraic function like this, we follow two core steps:

• Solve for \( x \) in the equation \( y = f(x) \). For \( y = 2x + 9 \), subtract 9 from both sides and then divide by 2 to isolate \( x \).
• Exchange \( x \) and \( y \) to express the inverse, giving us \( f^{-1}(x) = \frac{x - 9}{2} \).
  

Both these methods showcase how algebraic manipulations help us find the inverse and understand the relationship fully.

Inverse Relationship Verification

The inverse relationship between two functions is a fundamental concept. It verifies that one function "undoes" the effect of the other. To establish that two functions \( f \) and \( f^{-1} \) are inverses, we perform specific verifications involving function composition.
By computing both \( f(f^{-1}(x)) \) and \( f^{-1}(f(x)) \) and checking whether each equals \( x \), we ascertain their inverse nature. In our worked example:

• For \( f(f^{-1}(x)) = f\left(\frac{x-9}{2}\right) = 2\left(\frac{x-9}{2}\right) + 9 = x \), we verify that applying \( f \) after \( f^{-1} \) returns the input \( x \).
• Similarly, \( f^{-1}\left(f(x)\right) = \frac{2x+9-9}{2} = \frac{2x}{2} = x \) ensures \( f^{-1} \) followed by \( f \) also results in \( x \).

This thorough verification demonstrates how two functions truly are inverses, offering a clear understanding of their inverse relationship.