Problem 5 - 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. $$ f(x)=5 x-9 \text { and } g(x)=\frac{x+5}{9} $$

Step-by-Step Solution

Answer

The composition of functions \(f(g(x))\) and \(g(f(x))\) both equal to \(x\). This implies that the functions \(f\) and \(g\) are inverses of each other.

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

To substitute \(g(x)\) into \(f(x)\), use \(g(x) = \frac{x+5}{9}\) and substitute it into \(f(x) = 5x-9\) to get \(f(g(x)) = 5*\(\frac{x+5}{9}\)-9\). Now simplify this expression.

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

Simplify the expression to get \(f(g(x)) = \frac{5*(x+5)}{9}-9\). With further simplification this equals \(x-4\).

Step 3: Find \(g(f(x))\)

Begin by substituting the equation \(f(x)=5x-9\) into \(g(x)\). So \(g(f(x)) = \frac{5x-9+5}{9}\). Now simplify that expression.

Step 4: Simplify \(g(f(x))\)

After simplification, \(g(f(x)) = \frac{5x-4}{9}\). If we go on to simplify it further, it equals \(x\).

Step 5: Make the conclusion

Since \(f(g(x)) = x\) and \(g(f(x)) = x\), this means that the two functions \(f\) and \(g\) are inverses of each other. This is because by definition, two functions are inverses of each other if and only if \(f(g(x)) = x\) and \(g(f(x)) = x\) for all \(x\) in the domain of \(x\).

Key Concepts

Composition of Functions

Understanding the composition of functions is key when working with two functions like \(f\) and \(g\). To compose functions, you substitute one function into another. In our case, this means finding \(f(g(x))\) and \(g(f(x))\).

**Steps to Compose Functions**

• Identify the inner function to plug into the outer function. Here, you substitute \(g(x) = \frac{x+5}{9}\) into \(f(x) = 5x-9\) to find \(f(g(x))\).
• Repeat the process with \(f(x)\) as the inner function and substitute it into \(g(x)\) to find \(g(f(x))\).

Composing functions helps us see how one function affects another and is crucial in checking if two functions are inverses.

Function Notation

Function notation is a way to show what's happening with functions clearly. It uses symbols like \(f(x)\) or \(g(x)\) to represent function relationships. This notation helps in understanding what input goes into what function.

**Important Points on Function Notation**

• \(f(x)\) means "the function \(f\) evaluated at \(x\)." It's like getting an output after putting an input \(x\) through the function \(f\).
• When we see \(f(g(x))\), it means we first evaluate the function \(g\) at \(x\), and then use that result as the input for \(f\).
• This notation keeps track of what operations you're doing, making it easier to follow and solve problems.

Mastering function notation lets you clearly express and solve function-related problems with confidence.

Algebraic Simplification

Simplifying expressions is a fundamental part of algebra. When combining functions, you'll often simplify to see if the functions are inverses. Here's how simplification played a role in our exercise:

**Steps to Simplify Function Compositions**

• For \(f(g(x)) = 5\times\frac{x+5}{9} - 9\), distribute the 5 across the terms inside the parentheses: \(\frac{5(x+5)}{9} - 9\).
• Simplify further by breaking down and combining like terms. This led us to \(x-4\).
• Reapply these steps for \(g(f(x)) = \frac{5x-9+5}{9}\), which simplifies to \(x\).

Simplifying these expressions allowed us to confirm that \(f(g(x)) = x\) and \(g(f(x)) = x\), proving that \(f\) and \(g\) are indeed inverses.

Effective algebraic simplification clarifies complex expressions and reveals the underlying relationships between functions.