Problem 1 - 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. $$ f(x)=4 x \text { and } g(x)=\frac{x}{4} $$

Step-by-Step Solution

Answer

The function \(f(x) = 4x\) and \(g(x) = \frac{x}{4}\) are inverses of each other, as both \(f(g(x))\) and \(g(f(x))\) equate to \(x\).

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

To find the composite function \(f(g(x))\), substitute \(g(x)\) into \(f(x)\). In our case, \(g(x)\) is \(\frac{x}{4}\), so we substitute this into \(f(x) = 4x\) to get \(f(g(x)) = 4[\frac{x}{4}] = x\).

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

To find the composite function \(g(f(x))\), substitute \(f(x)\) into \(g(x)\). Here, \(f(x)\) is \(4x\), so we substitute this into \(g(x) = \frac{x}{4}\) to yield \(g(f(x)) = \frac{4x}{4} = x\).

Step 3: Check for inverse

The functions \(f(x)\) and \(g(x)\) are inverses of each other if \(f(g(x)) = x\) and \(g(f(x)) = x\). In our results, we see that both \(f(g(x))\) and \(g(f(x))\) equal to \(x\), so we can conclude that these functions are indeed inverses.

Key Concepts

Composite Functions

A composite function is created when one function is applied to the result of another function. Take two functions, say \(f(x)\) and \(g(x)\). The composite function \(f(g(x))\) means you first apply \(g\) to \(x\) and then apply \(f\) to the result of \(g(x)\). The order of functions matters significantly. When calculating composite functions:

• Substitute the entire output of one function into the other.
• Remember that \(f(g(x))\) might not always be the same as \(g(f(x))\).

In the given exercise, substituting \(g(x) = \frac{x}{4}\) into \(f(x) = 4x\) results in \(f(g(x)) = x\). This process simplifies because the functions are constructed to perfectly reverse each other's operations. Each operation cancels out the other, leaving us with \(x\).

Function Operations

Function operations are ways of combining or altering functions using addition, subtraction, multiplication, division, or composition. In our exercise, composition is key. But how does it work, and why?

• Composition applies one function to the result of another, effectively chaining their operations.
• This chaining creates complex operations from simpler building blocks.

For instance, illustrating with \(g(f(x))\), substitute \(f(x) = 4x\) into \(g(x) = \frac{x}{4}\). This yields \(g(f(x)) = x\), just as with \(f(g(x)) = x\). These compositions are transforming \(x\) through functions in such a way that they 'undo' each other, efficiently returning us back to the starting point.

Algebraic Functions

Algebraic functions are a type of function that involve basic operations like addition, subtraction, multiplication, and division, as well as composition. They are expressed using algebraic expressions. Understanding these is critical to grasping more complex algebra processes. Here are a few pointers:

• Algebraic expressions model real-world situations in straightforward mathematical terms.
• In this exercise, \(f(x) = 4x\) and \(g(x) = \frac{x}{4}\) represent these operations.

Notice how algebraic functions let us "undo" an operation by performing its inverse: multiplying by 4 and then dividing by 4 brings you back to your original number \(x\). This flip-flop nature helps us understand relations like inversion and symmetries in algebra.

College Algebra

College algebra extends high school algebra by diving deeper into more abstract or complex algebraic concepts. It includes topics like composite functions, inverse functions, and function operations. Often, understanding these principles relies on:

• Digging into concepts such as inverse relationships and compositions.
• Seeing the connections between different algebraic operations and their real-world implications.

The example from the exercise provides a glimpse into college algebra through its emphasis on inverse functions. By identifying operations that cancel each other, students practice analyzing functions at a higher level. Recognizing that both \(f(g(x)) = x\) and \(g(f(x)) = x\) indicates their inverse nature, helping build a foundation for further studies in mathematics.