Problem 96 - Applied Calculus

Question

If \(f(x)=a x,\) then \(f(f(x))=?\)

Step-by-Step Solution

Answer

\( f(f(x)) = a^2x \).

Step 1: Understand the function

The function given is a linear function, defined as \( f(x) = ax \). This means for any input \( x \), the function outputs \( ax \).

Step 2: Apply the function once

Apply the function \( f(x) \) to \( x \) to find \( f(x) \) itself. This results in: \( f(x) = ax \).

Step 3: Apply the function to itself

Now we need to find \( f(f(x)) \). Substitute \( ax \) into the function: \( f(ax) = a(ax) = a^2x \).

Step 4: Conclude the result

Therefore, when the function is applied to itself, we get \( f(f(x)) = a^2x \).

Key Concepts

Linear Functions

A linear function is one of the simplest types of functions you can come across in algebra. It is typically represented in the form \( f(x) = mx + b \), where \( m \) and \( b \) are constants. However, in our specific case, the linear function is given as \( f(x) = ax \), where \( a \) is a constant, and \( b \) is zero.

Key characteristics of linear functions include:

• They create straight lines when graphed on a coordinate plane.
• The slope of the line is determined by the constant \( a \).
• As it is a proportional relationship, if \( a > 0 \), the function is increasing. If \( a < 0 \), it is decreasing.

Linear functions are foundational in algebra because they model relationships where one variable changes at a constant rate relative to another.

Algebra

Algebra involves using symbols and letters to represent numbers and quantities in formulas and equations. It is an essential branch of mathematics that allows us to solve problems in a structured way. In our example, we use algebra to manipulate and solve the function \( f(x) = ax \) to find \( f(f(x)) \).

Let's break down the process:

• Recognize that \( f(x) = ax \) means for any \( x \), the output is \( ax \).
• To find \( f(f(x)) \), substitute \( f(x) \) into itself: \( f(ax) = a(ax) = a^2x \).

Algebra enables us to perform operations systematically, ensuring that we handle the function composition correctly, resulting in an accurate solution for \( f(f(x)) \).

Mathematical Operations

Mathematical operations are the foundation of solving equations and performing calculations in algebra. In this exercise, we are primarily using multiplication, which is one of the basic operations.

Here's how mathematical operations come into play in our function composition:

• The initial function \( f(x) = ax \) involves multiplying \( x \) by \( a \).
• When applying \( f \) to itself, we substitute \( ax \) into the function, resulting in the operation \( a(ax) \), which simplifies to \( a^2x \).

Understanding how to handle operations like multiplication is crucial for performing function compositions correctly. Make sure to pay attention to the properties of operations, as this ensures precision in calculations and reinforces algebraic skills.