Problem 32 - A Graphical Approach to College Algebra

Question

If \(f(x)=x,\) then for any function \(g,(f \circ g)(x)=\) \((g \circ f)(x)=\)______.

Step-by-Step Solution

Answer

Both \((f \circ g)(x)\) and \((g \circ f)(x)\) are equal to \(g(x)\).

Step 1: Understand the Function Composition

Function composition involves applying one function to the result of another. The notation \((f \circ g)(x)\) means you apply \(g(x)\) first, then \(f(x)\) on the result, and \((g \circ f)(x)\) means apply \(f(x)\) first, then \(g(x)\) on that result.

Step 2: Substitute in \(f(x)=x\) for \((f \circ g)(x)\)

Since \(f(x) = x\), when you apply \(f\) to \(g(x)\), it simply returns \(g(x)\) itself. Thus, \((f \circ g)(x) = f(g(x)) = g(x)\).

Step 3: Substitute in \(f(x)=x\) for \((g \circ f)(x)\)

For the expression \((g \circ f)(x)\), where you apply \(f(x)\) first, you substitute \(x\) into \(g\). Therefore, \((g \circ f)(x) = g(f(x)) = g(x)\).

Key Concepts

Identity Function

The identity function is one of the simplest functions you will encounter in mathematics. It is defined as a function that returns the input as the output. In mathematical terms, it is written as \( f(x) = x \). This means that whatever value you give to the function, you get the same value back.

The identity function acts like a neutral element in function composition. When you compose any function with the identity function—whether it's added before or after another function—the result remains the original function.

• For example, if you have a function \( g(x) \), composing it with \( f(x) = x \) results in \( g(x) \) again.
• In our exercise, both \( (f \circ g)(x) \) and \( (g \circ f)(x) \) simplify to \( g(x) \) because the identity function does not alter the output of \( g(x) \).

Understanding the role of the identity function helps simplify complex problems in function composition, ensuring you know when your function calculations will remain unchanged.

Function Notation

Function notation serves as a compact way to express mathematical functions. It's universally used across mathematics to clarify which function you're dealing with and what variable it depends on. At the core, function notation is written as \( f(x) \), where \( f \) denotes the function's name or label, and \( x \) represents the input variable.

Using function notation has several benefits:

• It helps to keep track of different functions, distinguishing them by labels like \( f(x) \), \( g(x) \), etc.
• It allows for the clear expression of composition, for instance, \( (f \circ g)(x) \), so you know which function is applied first.
• It simplifies the representation of the operations performed, such as adding, subtracting, or composing functions.

Function notation is essential for any work involving functions, as it provides a precise language to discuss and reason about how different functions interact through operations like substitution and composition.

Substitution in Functions

Substitution in functions is a fundamental operation where you replace the input variable of a function with another expression or function. This is a critical step in working with function compositions. By substituting specific values or expressions, we can evaluate function compositions or solve equations involving multiple functions.

In our exercise:

• When substituting for \( (f \circ g)(x) \), you replace \( g(x) \) using the identity function \( f(x) = x \), resulting in \( g(x) \).
• Similarly, for \( (g \circ f)(x) \), you use swap \( f(x) \) with \( x \), which means \( g(f(x)) = g(x) \).

Substitution ensures you follow the order of operations correctly in complex expressions, and it preserves the integrity of function interaction. Grasping substitution, along with understanding identity functions and function notation, helps deconstruct difficult problems into straightforward steps.