Problem 37 - A Graphical Approach to Precalculus with Limits

Question

Use the definition of inverse functions to show analytically that \(f\) and \(g\) are inverses. $$f(x)=3 x-7, \quad g(x)=\frac{x+7}{3}$$

Step-by-Step Solution

Answer

Functions \( f(x) = 3x - 7 \) and \( g(x) = \frac{x+7}{3} \) are inverses because \( f(g(x)) = x \) and \( g(f(x)) = x \).

Step 1: Understand the Definition of Inverse Functions

Two functions, \( f \) and \( g \), are inverses of each other if their compositions, \( f(g(x)) \) and \( g(f(x)) \), both equal \( x \) for all \( x \) in the domain of the inside function.

Step 2: Compute the Composition \( f(g(x)) \)

Substitute \( g(x) = \frac{x+7}{3} \) into \( f(x) = 3x - 7 \). The expression becomes \( f(g(x)) = 3\left(\frac{x+7}{3}\right) - 7 \). Simplify this expression: \[ f(g(x)) = 3 \times \frac{x+7}{3} - 7 = (x + 7) - 7 = x\]

Step 3: Compute the Composition \( g(f(x)) \)

Substitute \( f(x) = 3x - 7 \) into \( g(x) = \frac{x+7}{3} \). The expression becomes \( g(f(x)) = \frac{(3x - 7) + 7}{3} \). Simplify this expression:\[g(f(x)) = \frac{3x - 7 + 7}{3} = \frac{3x}{3} = x\]

Step 4: Conclude That \( f \) and \( g \) Are Inverses

Since both compositions \( f(g(x)) = x \) and \( g(f(x)) = x \) hold true, by the definition of inverse functions, we can conclude that \( f(x) = 3x - 7 \) and \( g(x) = \frac{x+7}{3} \) are inverses of each other.

Key Concepts

Compositions of Functions

One of the essential concepts in understanding inverse functions is the idea of function composition. Basically, when you combine two functions, say \( f \) and \( g \), you're constructing a new function by applying one function to the result of another. Mathematically, composition is expressed as \( f(g(x)) \) or \( g(f(x)) \).When working with inverses, we want to show that these compositions simplify to \( x \). Here's why:

• When you compose \( f \) and \( g \), you're essentially performing an operation and then undoing it, returning to your starting point, \( x \).
• Confirming \( f(g(x)) = x \) and \( g(f(x)) = x \) is key to proving that two functions are inverses.

In the exercise, showing that \( f(g(x)) = x \) and \( g(f(x)) = x \) confirms they undo each other's operations. This is like having a lock and its corresponding key.

Function Inverses

Inverse functions are like mathematical undo buttons. If \( f \) is a function that transforms \( x \) into a different value, the inverse function, often denoted as \( f^{-1} \), reverses this transformation, bringing us back to \( x \).A function \( f \) will have an inverse \( g \) if:

• Both compositions \( f(g(x)) \) and \( g(f(x)) \) simplify to \( x \).
• This property indicates that \( f \) and \( g \) mutually reverse each other's actions.

In our example, we successfully demonstrated that both compositions simplify to \( x \):

• \( f(g(x)) = 3\left(\frac{x+7}{3}\right) - 7 = x + 7 - 7 = x \)
• \( g(f(x)) = \frac{3x - 7 + 7}{3} = \frac{3x}{3} = x \)

Through these calculations, \( f(x) = 3x - 7 \) and \( g(x) = \frac{x+7}{3} \) are proven to be inverses.

Algebraic Simplification

Algebraic simplification is a crucial step in solving and verifying function inverses. This process involves reducing expressions to their simplest forms, which helps verify that functions truly reverse each other's effects.Here's a breakdown of the simplification process:

• For \( f(g(x)) \), substitute \( g(x) \) into \( f(x) \) and simplify: \( f(g(x)) = 3\left(\frac{x+7}{3}\right) - 7 = x + 7 - 7 = x \).
• For \( g(f(x)) \), substitute \( f(x) \) into \( g(x) \) and simplify: \( g(f(x)) = \frac{3x - 7 + 7}{3} = \frac{3x}{3} = x \).

Each composition simplifies down to \( x \), confirming that the operations of \( f \) and \( g \) perfectly undo one another. This step-by-step reduction process helps solidify your understanding of inverse functions and their properties.