Problem 29 - A Graphical Approach to Precalculus with Limits

Question

Answer each of the following. If \(f\) and \(g\) are inverses, then \((f \circ g)(x)=\) __________ and __________$$=x$$.

Step-by-Step Solution

Answer

(f \circ g)(x) = x and (g \circ f)(x) = x.

Step 1: Understanding Inverse Functions

Two functions \(f\) and \(g\) are inverses of each other if \(f(g(x)) = x\) and \(g(f(x)) = x\) for all \(x\) in the domain of \(g\) and \(f\) respectively. These two equations describe how the functions \(f\) and \(g\) 'undo' each other.

Step 2: Applying the Inverse Concept to Composition

Since \(f\) and \(g\) are inverses, the definition tells us that \((f \circ g)(x) = f(g(x)) = x\). This reflects that applying \(g\) and then \(f\) returns the original input \(x\).

Step 3: Completing the Expression Using Inverses

Besides the expression \((f \circ g)(x)=x\), for inverses, the other composition is \((g \circ f)(x) = g(f(x)) = x\). This means applying \(f\) and then \(g\) also returns the original input \(x\).

Key Concepts

Function Composition

Function composition is a way to combine two functions, say \( f \) and \( g \), into a single function. This happens when the output of one function becomes the input of the other. Function composition is denoted by \((f \circ g)(x) = f(g(x))\).

• Steps: First you apply the function \( g \) to \( x \), giving you \( g(x) \).
• Next, you use this result as the input for the function \( f \), yielding \( f(g(x)) \).

This method allows you to layer operations and see how functions interact with each other.
For a practical example, if \( f(x)=2x \) and \( g(x)=x+3 \), then composing these would give \((f \circ g)(x) = f(g(x)) = 2(x+3) = 2x + 6\). Consequently, understanding composition is crucial for exploring inverse functions.

Inverse Relations

Inverse relations between two functions occur when their compositions return the original input for every element in their domains. Specifically, for functions \( f \) and \( g \) to be inverses, the criterion is:\( f(g(x)) = x \) and \( g(f(x)) = x \).
This symmetric property denotes that applying one function after the other undoes whatever the first did, bringing you full-circle back to your starting point.

• Usefulness: This attribute helps in many branches of mathematics as it identifies when two functions can exactly "cancel" each other.
• A common example includes trigonometric functions like the sine and arcsine functions, with \( \sin(\arcsin(x)) = x \) and \( \arcsin(\sin(x)) = x \).

Realizing when two functions are inverses facilitates problem-solving and reveals underlying patterns within mathematics.

Domain and Range Concepts

To fully understand inverse functions, grasping domain and range is essential.

• The domain of a function contains all of the possible input values,
• The range encompasses all possible outputs.

When discussing inverse functions, both domain and range need attention.
* For \( f \) and \( g \) to be inverses, the domain of \( f \) must match the range of \( g \), while the domain of \( g \) should match the range of \( f \). This ensures that every input and output pair correctly reverses.
Consider the functions \( f(x) = x^2 \) with domain \( x \geq 0 \), and \( g(x) = \sqrt{x} \), their ranges and domains align perfectly to make them inverse functions.
Hence, evaluating these aspects of functions not only solidifies inverse understanding but also ensures mathematical accuracy when finding or confirming function pairs.