Problem 199 - Calculus Volume 1

Question

Use composition to determine which pairs of functions are inverses. \(f(x)=8 x, g(x)=\frac{x}{8}\)

Step-by-Step Solution

Answer

\((f(x)=8x)\) and \(g(x)=x/8\) are inverses.

Step 1: Understand Inverse Functions

Two functions are inverses of each other if their composition results in the identity function, meaning for functions \(f\) and \(g\), \(f(g(x)) = x\) and \(g(f(x)) = x\) for all \(x\) in their domains.

Step 2: Compose f and g

Calculate the composition \(f(g(x))\): \[ f(g(x)) = f\left(\frac{x}{8}\right) = 8 \cdot \frac{x}{8} = x \] This shows that when \(g(x)\) is substituted into \(f(x)\), the result simplifies to \(x\).

Step 3: Compose g and f

Now calculate the composition \(g(f(x))\): \[ g(f(x)) = g(8x) = \frac{8x}{8} = x \] This shows that when \(f(x)\) is substituted into \(g(x)\), the result also simplifies to \(x\).

Step 4: Conclude the Inverse Relationship

Since both \(f(g(x)) = x\) and \(g(f(x)) = x\), we conclude that \(f(x)\) and \(g(x)\) are inverses of each other.

Key Concepts

Function Composition

Function composition is a key operation in mathematics where the output of one function becomes the input of another function. This operation is denoted as

• \( (f \circ g)(x) = f(g(x)) \),

meaning you apply function \( g \) to \( x \), and then function \( f \) to the result of \( g(x) \).

Using function composition, we can verify the relationship between two functions, particularly to determine if they are inverses. In the provided exercise, we composed the functions \( f(x) = 8x \) and \( g(x) = \frac{x}{8} \).

The order of composition is important, as shown:

• Composing \( f(g(x)) \): Insert \( g(x) = \frac{x}{8} \) into \( f(x) \), resulting in \( f(g(x)) = f\left( \frac{x}{8} \right) = x \).
• Composing \( g(f(x)) \): Insert \( f(x) = 8x \) into \( g(x) \), resulting in \( g(f(x)) = g(8x) = x \).

This simplification to \( x \) in both cases signals that these functions are indeed inverses of each other.

Identity Function

In mathematics, the concept of the identity function is quite straightforward but plays a crucial role in understanding function inverses. An identity function, usually denoted as \( I(x) \), is a function that maps any element \( x \) to itself, such as \( I(x) = x \).

When determining if two functions are inverses, the goal is to see if their compositions yield this identity function.

• For functions \( f \) and \( g \) to be inverses, \( f(g(x)) = x \) and \( g(f(x)) = x \) must hold true for all values in their domains.

In simpler terms, applying one function and then the other returns the original input.

In our exercise, when both compositions of \( f \) and \( g \) result in \( x \), we confirm that \( f(x) = 8x \) and \( g(x) = \frac{x}{8} \) are indeed inverse functions because they generate the identity function.

Domain of a Function

The domain of a function is the complete set of possible input values (or \( x \)-values) for which the function is defined. Understanding domains is essential when dealing with function compositions and inverses.

For example, in the given functions \( f(x) = 8x \) and \( g(x) = \frac{x}{8}\), it is important to ensure they can actually be applied to these inputs without leading to undefined operations, such as division by zero.

• The domain of \( f(x) = 8x \) includes all real numbers because the operation of multiplying by 8 is always valid.
• The domain of \( g(x) = \frac{x}{8} \) also includes all real numbers, provided that \( x \) is real, since \( x \) divided by 8 is always possible.

Considering these domains ensures both compositions, \( f(g(x)) \) and \( g(f(x)) \), are valid throughout, confirming the functions are true inverses.