Problem 4 - 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. $$ f(x)=4 x+9 \text { and } g(x)=\frac{x-9}{4} $$

Step-by-Step Solution

Answer

The functions \(f(x)=4x+9\) and \(g(x)=\frac{x-9}{4}\) are inverses because both \(f(g(x))\) and \(g(f(x))\) equal to \(x\).

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

To calculate \(f(g(x))\), it's necessary to substitute \(g(x)\) into \(f(x)\). Based on the given functions, this would become \(f\left(\frac{x-9}{4}\right)=4\left(\frac{x-9}{4}\right) + 9\). Simplifying this results in \(f(g(x))=x\).

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

For \(g(f(x))\), the function \(f(x)\) should be substituted into \(g(x)\). So, \(g(4x+9)=\frac{4x+9-9}{4}\). After simplifying, the result is \(g(f(x))=x\).

Step 3: Determine if functions are inverses

To determine if \(f\) and \(g\) are inverses, \(f(g(x))\) and \(g(f(x))\) should equal \(x\). As the results of both \(f(g(x))\) and \(g(f(x))\) are \(x\), it can be concluded that \(f\) and \(g\) are indeed inverses.

Key Concepts

Composition of Functions

When studying functions, an important operation is the composition of functions. This involves taking two functions and combining them to form a new function. The composition of two functions, say \( f(x) \) and \( g(x) \), is denoted by \( f(g(x)) \) or \( g(f(x)) \), depending on the order of application. The order is crucial, as composition is not necessarily commutative, meaning \( f(g(x)) \) may not equal \( g(f(x)) \).

To compose functions, we substitute the entire second function into the first function wherever we see the input variable. For example, if we have \( f(x) = 4x + 9 \) and \( g(x) = (x - 9)/4 \), then to find \( f(g(x)) \), we replace the \( x \) in \( f(x) \) with \( g(x) \). This process is a form of function transformation where one function is nested within another, helping students understand how multiple functions can be interconnected.

Inverse Functions

Inverse functions are a pair of functions that reverse each other's actions. If \( f(x) \) takes an input and transforms it into an output, then an inverse function, denoted as \( f^{-1}(x) \), starts with the output and recovers the original input. For two functions to be inverses of each other, the composition \( f(g(x)) \) and \( g(f(x)) \) should both result in \( x \), the original input.

This is a powerful concept because it allows us to 'undo' the effect of a function, thereby solving for the original value. When we found that \( f(g(x)) = x \) and \( g(f(x)) = x \) with our given functions, it demonstrated that they are perfect inverses—a critical insight when tackling problems in various fields like algebra and calculus.

Algebraic Substitution

Implementing Algebraic Substitution

In solving equations and functions, algebraic substitution is a key tool that simplifies complex expressions. It involves replacing a variable with its equivalent expression from another equation or function. This is exactly what we do when finding the composition of functions. For instance, substituting \( f(x) \) with \( 4x + 9 \) into \( g(x) \) gives us the ability to express one function in terms of another, facilitating the simplification process. Mastering the execution of algebraic substitution can significantly help students navigate more advanced topics in algebra, where manipulation of equations becomes more intricate.

Function Simplification

Function simplification involves reducing a function to its most basic form without changing its value or properties. It often includes arithmetic operations like distributing multiplicative factors, combining like terms, and canceling common factors in fractions. For example, when we calculated \( g(4x + 9) \), part of the simplification process involved canceling the '+9' and '-9'. Simplifying functions makes them easier to work with, especially when graphing or applying further mathematical procedures. In algebra, simplification is a precursor to higher-level problem solving and helps in understanding function behaviours more intuitively.