Problem 203 - Calculus Volume 1

Question

For the following exercises, use composition to determine which pairs of functions are inverses. $$ f(x)=\frac{1}{x-1}, x \neq 1, g(x)=\frac{1}{x}+1, x \neq 0 $$

Step-by-Step Solution

Answer

The functions are inverses.

Step 1: Understand the composition definition

Two functions \( f \) and \( g \) are inverses if \( f(g(x)) = x \) and \( g(f(x)) = x \) for all \( x \) in the domain of each composite function.

Step 2: Compose \( f(g(x)) \)

To find \( f(g(x)) \), substitute \( g(x) = \frac{1}{x} + 1 \) into \( f(x) \). This means substitute \( \frac{1}{x}+1 \) for \( x \) in \( f(x) \):\[f(g(x)) = f\left(\frac{1}{x}+1\right) = \frac{1}{\left(\frac{1}{x}+1\right) - 1}\]Simplify the expression:\[f(g(x)) = \frac{1}{\frac{1}{x}} = x\]

Step 3: Compose \( g(f(x)) \)

To find \( g(f(x)) \), substitute \( f(x) = \frac{1}{x-1} \) into \( g(x) \). This means substitute \( \frac{1}{x-1} \) for \( x \) in \( g(x) \):\[g(f(x)) = g\left(\frac{1}{x-1}\right) = \frac{1}{\left(\frac{1}{x-1}\right)} + 1 = (x-1) + 1\]Simplify the expression:\[g(f(x)) = x\]

Step 4: Conclusion about function inverses

Since both compositions, \( f(g(x)) \) and \( g(f(x)) \), simplify to \( x \), the functions \( f(x) \) and \( g(x) \) are inverses of each other as their compositions yield the identity function.

Key Concepts

Composition of Functions

When we talk about the composition of functions, we are referring to the process where one function is applied to the result of another function. Think of it as a function inside a function, much like layers on a cake. In simpler terms, if you have two functions, say \( f(x) \) and \( g(x) \), the composition \( f(g(x)) \) involves inserting \( g(x) \) into \( f(x) \).
To perform this, take the output of \( g(x) \) and substitute it as the input for \( f(x) \). Symbolically, it’s represented as \( (f \circ g)(x) = f(g(x)) \) and this means you do \( g(x) \) first and then apply \( f \).

• For example, if \( f(x) = 2x \) and \( g(x) = x+3 \), then \( f(g(x)) = f(x+3) = 2(x+3) \).
• This simplifies to \( 2x + 6 \).

This process helps determine important relationships between functions, such as inverting one another.

Function Inverses

Functions are inverse of each other when they "undo" each other’s actions. Simply put, applying one function and then its inverse will bring you back to your starting point. For example, if a function takes an input \( x \) and increases it, its inverse will take the result and decrease it back to \( x \).
Mathematically, two functions \( f(x) \) and \( g(x) \) are inverses if both \( f(g(x)) = x \) and \( g(f(x)) = x \). They should return the input back to its original value through both compositions. Hence, to show that two functions are inverses, check the result of both compositions.

• To test if \( f(x) \) and \( g(x) \) are inverses, calculate \( f(g(x)) \) and \( g(f(x)) \).
• If both equal \( x \), then \( f(x) \) and \( g(x) \) are inverses.

Finding inverse functions helps solve equations and understand function behaviors.

Composite Function Domain

The domain of a composite function refers to all the possible input values (\( x \)) for which the composition \( f(g(x)) \) is defined. It involves ensuring that both individual functions must have valid outputs and inputs.
When finding the domain of a composite function like \( f(g(x)) \), determine:

• The domain of \( g(x) \), because \( g(x) \) must be valid to produce outputs for \( f \).
• The range of \( g(x) \), ensuring each output can function as an input for \( f \).

This cross-check ensures that every part of the composite function works smoothly. Without valid input and output ranges, the composition would be undefined at certain points. For instance, in our solution, knowing \( g(x) = \frac{1}{x} + 1 \) means \( x eq 0 \) and \( f(x) = \frac{1}{x-1} \) means \( x eq 1 \), giving a combined restriction for a valid computation.