Problem 28 - 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

Both conditions are satisfied, so \( f \) and \( g \) are inverses of each other.

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

First, we need to substitute the function \( f(x) \) into the function \( g \). This means we want to calculate \( g(f(x)) \) and check if it equals \( x \).Given \( f(x) = \frac{3-x}{4} \), substitute it into \( g(x) = 3 - 4x \):\[g(f(x)) = 3 - 4\left(\frac{3-x}{4}\right)\]Simplify: \[g(f(x)) = 3 - 4 \cdot \frac{3}{4} + 4 \cdot \frac{x}{4}= 3 - 3 + x = x\]Since \( g(f(x)) = x \), the first condition is satisfied.

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

Now, substitute \( g(x) \) into \( f \) to calculate \( f(g(x)) \) and check if it equals \( x \).Given \( g(x) = 3 - 4x \), substitute it into \( f(x) = \frac{3-x}{4} \):\[f(g(x)) = \frac{3 - (3 - 4x)}{4}\]Simplify:\[f(g(x)) = \frac{3 - 3 + 4x}{4} = \frac{4x}{4} = x\]Since \( f(g(x)) = x \), the second condition is satisfied.

Key Concepts

Function Composition

Function composition is a crucial concept when working with two functions, especially when trying to determine if one is the inverse of the other. Simply put, function composition involves plugging one function into another. In the case of functions \(f\) and \(g\), we look at \(g(f(x))\), which means taking \(f(x)\) and using it as the input for \(g(x)\). Similarly, \(f(g(x))\) refers to using \(g(x)\) as the input of \(f(x)\).Here’s how it works:

• If \(f\) and \(g\) are inverses of each other, then \(g(f(x))\) should simplify to give you \(x\).
• Similarly, the result of \(f(g(x))\) should also simplify to \(x\).

Once these conditions are met, \(f\) and \(g\) can be confirmed as inverse functions. This is a neat concept, because it uses simple operations to check the relationship between two functions.

Algebraic Simplification

Algebraic simplification involves tidying up expressions so they become easier to understand and work with. It’s an important step in function composition for inverse functions.Let’s explore how:

• Starting with \(g(f(x)) = 3 - 4\left(\frac{3-x}{4}\right)\), we simplify by distributing and cancelling terms.
• The key is recognizing that operations like \(-4 \cdot \frac{3}{4}\) simplify to \(-3\), and \(-4 \cdot -\frac{x}{4}\) simplifies to \(+x\).

Through this method, we reduce complicated expressions into simpler forms that reveal whether the output is just \(x\). This verification confirms the composition connects back to the original input, affirming the inverse relationship between functions.

One-to-One Functions

One-to-One Functions, often labeled as injective, are a fundamental requirement for finding inverse functions. A function is one-to-one if every output is mapped from a unique input. In simple terms, different inputs always yield different outputs.Why is this important for inverses?

• If a function is not one-to-one, it cannot have a proper inverse because multiple inputs may produce the same output.
• In our case, verifying \(f\) and \(g\) return exactly \(x\) upon function composition checks this one-to-one character.

Thus, having this property guarantees that inverse functions will work correctly, linking each output precisely to a single input. Without this, an inverse would be impossible or not well-defined. Recognizing one-to-one behavior allows us to confidently explore inverse relationships in functions.