Problem 84 - College Algebra with Corequisite Support

Question

Let \(f(x)=\frac{1}{x}\). (a) Find \((f \circ f)(x)\). (b) Is \((f \circ f)(x)\) for any function \(f\) the same result as the answer to part (a) for any function? Explain.

Step-by-Step Solution

Answer

(f \circ f)(x) = x. No, only involutive functions generally satisfy this.

Step 1: Understanding Composition

The composition of two functions, \((f \circ g)(x) \) means we first apply the function \( g(x) \), and then apply the function \( f \) to the result of \( g(x) \). So, \((f \circ f)(x)\) means applying \( f(x) \) to itself, that is, substituting \( f(x) \) into \( f \).

Step 2: Substitute \( f(x) \) into itself

To find \((f \circ f)(x)\), substitute \( f(x) = \frac{1}{x} \) into \( f \). Thus, \( (f \circ f)(x) = f(f(x)) = f\left(\frac{1}{x}\right) \).

Step 3: Apply Function to Result

Apply \( f \) to \( \frac{1}{x} \): \( f\left(\frac{1}{x}\right) = \frac{1}{\frac{1}{x}} \). This simplifies to \( x \), because \( \frac{1}{\frac{1}{x}} = x \cdot \frac{1}{1} = x \). Thus, \((f \circ f)(x) = x \).

Step 4: Exploration of General Case

For a general function \( g(x) \), \((g \circ g)(x) \) is specific to the function used and often not equal to \( x \). Therefore, the result \((f \circ f)(x) = x\) is unique to functions where \( g(g(x)) = x \), often called an involution.

Key Concepts

Inverse Functions

An inverse function is a fundamental concept in mathematics that essentially reverses the effect of the original function. In simple terms, for a function \( f(x) \), the inverse function \( f^{-1}(x) \) satisfies the condition \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \). This means when you apply a function to a value and then use its inverse, you return to the original value you started with.

Not every function has an inverse. A function needs to be bijective, meaning it must be both injective and surjective, to have an inverse. Here's how these properties work:

• Injective (One-to-One): This ensures that different inputs always result in different outputs in a function.
• Surjective (Onto): Every element in the output set is covered by the function, meaning the function's output exactly matches its codomain.

When these conditions are satisfied, the function can be reversed by its inverse, which is crucial in many mathematical applications ranging from calculus to algebra.

Involution

Involution refers to a special type of function that is its own inverse. This means that applying the function twice results in the original value. Mathematically, a function \( f \) is an involution if it satisfies the property \( f(f(x)) = x \) for all \( x \) in its domain.

The function \( f(x) = \frac{1}{x} \) is a classic example of an involution, illustrated by how \( (f \circ f)(x) = x \). In the specific case where \( f(x) = \frac{1}{x} \), applying \( f \) twice leads us back to our starting point \( x \), making it an involution.

This property has intriguing applications and implications:

• Simplicity: Many algebraic problems simplify when involving involutions, as the backwards application instinctively undoes previous steps.
• Transformations: In graphics and data processing, involutions help revert transformations, enabling inverse operations like cryptography decryption.

Understanding involution is vital for students delving into more advanced topics of function theory.

Function Operations

Function operations involve processes like addition, subtraction, multiplication, and division of functions, but importantly also function composition. Function composition, \((f \circ g)(x)\), redirects the output of one function into another, much like a chain reaction where the outcome of \( g(x) \) becomes the input for \( f(x) \).

Let’s unwrap what we've seen in the math problem: The composition \((f \circ f)(x)\) indicates you apply the function \( f \) to its own output. Essentially, place \( f(x) \) instead of \( x \) in your equation.
This process might feel a bit like putting a puzzle together:

• First, determine \( f(x) \), here it's \( \frac{1}{x} \).
• Then apply the same function to this result: \( f(f(x)) = f(\frac{1}{x}) \).
• This simplifies to \( \frac{1}{\frac{1}{x}} = x \), completing the operation.

By studying function operations, students learn how to deftly manipulate functions like tools and master more intricate computations, ultimately gaining a powerful toolkit to tackle various mathematical challenges.