Problem 24 - Algebra and Trigonometry

Question

Use the Inverse Function Property to show that \(f\) and \(g\) are inverses of each other. \(f(x)=\frac{3-x}{4} ; \quad g(x)=3-4 x\)

Step-by-Step Solution

Answer

Functions are inverses because both compositions equal \( x \).

Step 1: Verify if \( f(g(x)) = x \)

First, substitute \( g(x) \) into \( f(x) \). That means we need to compute \( f(g(x)) = f(3-4x) \). Substitute 3-4x into f: \[ f(3-4x) = \frac{3 - (3-4x)}{4} \]. Simplify the expression: \( \frac{4x}{4} = x \). Thus, \( f(g(x)) = x \).

Step 2: Verify if \( g(f(x)) = x \)

Now, substitute \( f(x) \) into \( g(x) \). We need to compute \( g(f(x)) = g\left(\frac{3-x}{4}\right) \). Substitute \( \frac{3-x}{4} \) into \( g(x) \): \[ g\left( \frac{3-x}{4} \right) = 3 - 4\left(\frac{3-x}{4}\right) \]. Simplify to \( 3 - (3-x) = x \). Thus, \( g(f(x)) = x \).

Key Concepts

Function Composition

Function composition involves combining two functions so that the output of one function becomes the input of another. This concept is essential in verifying inverse functions.

• To find the composite function \( f(g(x)) \), you plug the entire function \( g(x) \) into the function \( f(x) \), replacing the variable \( x \) with \( g(x) \).
• Similarly, for \( g(f(x)) \), you substitute \( f(x) \) into the function \( g \).

In this exercise, we have:- First, evaluate \( f(g(x)) \) by substituting \( g(x) = 3 - 4x \) into \( f(x) = \frac{3-x}{4} \). It becomes \( f(3 - 4x) = \frac{3 - (3 - 4x)}{4} \).- Simplify the expression, and you get \( \frac{4x}{4} = x \).- This shows that \( f(g(x)) \) simplifies to \( x \), an important step in proving that these functions are inverses.

Algebraic Manipulation

Algebraic manipulation is the process of rearranging and simplifying expressions to reveal their underlying structure. In this context, it helps confirm whether two functions are inverses.

• First, identify the expressions where substitution is needed and carefully perform operations following the order of operations (PEMDAS: Parentheses, Exponents, Multiplication and Division, Addition and Subtraction.)
• After substitution, simplify the expressions step-by-step to reach a conclusion of \( x \). Simplifying usually involves performing basic arithmetic or using properties of division and multiplication.

For example, in verifying \( f(g(x)) \), we see:- \( f(g(x)) = \frac{3 - (3 - 4x)}{4} = \frac{4x}{4} = x \).- Similarly, for \( g(f(x)) = 3 - 4\left(\frac{3-x}{4}\right) \), it simplifies to \( 3 - (3-x) = x \).The ability to correctly manipulate and simplify expressions is key to verifying the inverse relationship.

Verification of Inverses

Verification of inverses means proving that two functions undo each other's operations, bringing you back to the original input, which is \( x \).

• An inverse function \( g(x) \) of \( f(x) \) satisfies the properties: \( f(g(x)) = x \) and \( g(f(x)) = x \).
• This implies both compositions must simplify to \( x \) for true inverse functions together.

In the given functions:- We confirmed \( f(g(x)) = x \) by substitution and simplification.- Similarly verified \( g(f(x)) = x \) also holds true, confirming the inverse nature.The critical takeaway is that for functions to be valid inverses, these compositions must both yield \( x \). The given step-by-step approach highlighted shows how a thorough verification process establishes this inverse relationship. Understanding this concept demonstrates how functions can precisely cancel each other out back to the original value of \( x \).