Problem 2 - College Algebra

Question

Find \(f(g(x))\) and \(g(f(x))\) and determine whether each pair of functions \(f\) and \(g\) are inverses of each other. $$ f(x)=6 x \text { and } g(x)=\frac{x}{6} $$

Step-by-Step Solution

Answer

The functions \(f\) and \(g\) are inverses of each other.

Step 1: Compute \(f(g(x))\)

Start by replacing \(g(x)\) in \(f(x)\) to find \(f(g(x))\). \(g(x)\) is given by \(\frac{x}{6}\). Thus, \(f(g(x))\) becomes:\\[f\left(\frac{x}{6}\right) = 6\left(\frac{x}{6}\right) = x\].

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

Next, replace \(f(x)\) in \(g(x)\) to find \(g(f(x))\). \(f(x)\) is determined as \(6x\). This makes \(g(f(x))\) equals to\\[g(6x) = \frac{6x}{6} = x\].

Step 3: Verify if \(f\) and \(g\) are Inverses

The functions \(f\) and \(g\) are inverses if \(f(g(x)) = g(f(x)) = x\). Since both \(f(g(x))\) and \(g(f(x))\) equal to \(x\), we can conclude that \(f\) and \(g\) are inverses of one another.

Key Concepts

Composition of Functions

Composition of functions involves combining two functions in a specific order to form a new function. When we compose functions, we use the variable from one function as the input to the other function. This is crucial for understanding how various functions interact and behave when combined. The composition is denoted by \(f(g(x))\), meaning we first apply the function \(g\) to \(x\), and then apply \(f\) to the result.
In our exercise, we have two functions: \(f(x) = 6x\) and \(g(x) = \frac{x}{6}\). By inserting \(g(x)\) into \(f\), we find \(f(g(x)) = 6\left(\frac{x}{6}\right)\), which simplifies to \(x\). Similarly, by placing \(f(x)\) into \(g\), we calculate \(g(f(x)) = \frac{6x}{6}\), also simplifying to \(x\).
This pattern shows a symmetric transformation between \(f\) and \(g\), where both operations return us to our original input \(x\). Such relationships in function composition are foundational for defining inversion.

Function Operations

Function operations refer to various methods of combining functions, including addition, subtraction, multiplication, division, and composition. Each type of operation yields a different result, but they all share the primary goal of building new functions from existing ones.
Composition is unique among these operations because it involves applying one function to the result of another. It effectively "chains" the functions together in a sequence, which can reveal important properties about how they interact. When done correctly, composition acts almost like solving a puzzle without changing the original information.
In this exercise, demonstrating function operations through composition shows how \(f(x)\) and \(g(x)\) interact to reverse, or "undo" one another, which is essential for discovering inverse functions.

Verifying Inverses

Verifying inverse functions means determining if two given functions, when composed with each other in both orders, return the initial input value \(x\). This is a critical step in establishing that two functions are indeed inverses.
To verify inverses, we need to compute both \(f(g(x))\) and \(g(f(x))\). If both expressions simplify to \(x\), then \(f\) and \(g\) are true inverses of each other.
In our example, both \(f(g(x)) = x\) and \(g(f(x)) = x\) confirm that \(f(x) = 6x\) and \(g(x) = \frac{x}{6}\) are inverses. This is because each function undo the effect of the other, returning to the original \(x\). This property of reversibility is fundamental to understanding inverses and has wide applications across mathematics. Understanding how to verify inverses helps in recognizing such pairings in different problem contexts.