Problem 32 - A Graphical Approach to Precalculus with Limits

Question

Answer each of the following. If \(f(x)=x,\) then for any function \(g\) , $$(f \circ g)(x)=(g \circ f)(x)=$$ __________.

Step-by-Step Solution

Answer

Both equal \(g(x)\).

Step 1 - Understanding Composition of Functions

Before we begin, recall that the composition of two functions, say \((f \circ g)(x)\), means applying function \(g\) to \(x\) first, and then applying function \(f\) to the result. Similarly, \((g \circ f)(x)\) is applying \(f\) to \(x\) first, and then applying \(g\) to the result.

Step 2 - Evaluate \((f \circ g)(x)\)

Since \(f(x) = x\), when composing \(f\) with \(g\), we start with \(g(x)\) and then apply \(f\), which simply returns the input. Therefore, \((f \circ g)(x) = f(g(x)) = g(x)\).

Step 3 - Evaluate \((g \circ f)(x)\)

Similarly, for \((g \circ f)(x)\), we first apply \(f\) to \(x\), which gives \(f(x) = x\), and then apply \(g\) to the result. Hence, \(g(f(x)) = g(x)\).

Step 4 - Conclude with Equality

By comparing both compositions, we have \((f \circ g)(x) = g(x)\) and \((g \circ f)(x) = g(x)\). Therefore, they both equal \(g(x)\).

Key Concepts

Identity Function

The identity function is a fundamental concept in mathematics. It's one of the simplest functions you can find. Defined as \( f(x) = x \), the identity function returns whatever value is inputted into it.
The identity function acts like a mirror: it reflects the input value back as output, unchanged. Here are some key properties:

• The identity function leaves any input unchanged. If you plug in 5, you get 5 out. If you plug in \( x \), you get \( x \) back.
• In function notation, it is denoted by \( I(x) \), where \( I(x) = x \).
• It's often used to simplify expressions and prove identities in calculus and algebra.

The identity function has a crucial role in function composition, as it does not alter other functions it is composed with.

Mathematical Functions

Mathematical functions are rules that assign each input exactly one output. Functions describe relationships between variables and are widely used in various fields of mathematics.
A function can be written as \( f(x) \), representing any process that changes input \( x \) to output by some rule. Features of functions include:

• Domains and Ranges: The domain is the set of all possible inputs, while the range is the set of possible outputs.
• Types: Functions can be linear, quadratic, polynomial, exponential, etc.
• Graphs: Functions are often represented graphically, showing the relationship between input and output visually.

Understanding these basics aids in analyzing complex functions and utilizing composition effectively, as seen when combining functions.

Function Evaluation

Evaluating a function involves substituting a specific value into the function's formula. For example, to find \( f(3) \) for the function \( f(x) = x^2 \), you simply substitute 3 for \( x \) to get \( f(3) = 9 \).
Function evaluation is straightforward but essential. It is critical when composing functions:

• Start by substituting the inner function's output into the outer function's equation.
• This process is repeated step-by-step to ensure accuracy.
• In exercises involving composition, always check each function's role carefully.

In the solved exercise, understanding the evaluation of each composition helped conclude that \((f \circ g)(x)\) and \((g \circ f)(x)\) both simplify to \(g(x)\). This demonstrates how basic evaluation techniques support more complex operations.