Problem 99 - Brief Applied Calculus

Question

$$ \text { If } f(x)=x+a \text { , then } f(f(x))=\text { ? } $$

Step-by-Step Solution

Answer

\( f(f(x)) = x + 2a \)

Step 1: Understand the Problem

We are given a function \( f(x) = x + a \) and we need to find \( f(f(x)) \). This means we must apply the function \( f \) twice in succession.

Step 2: Apply the Function to Itself

First, compute \( f(x) = x + a \). Then, we apply the function again to the result, so we need to calculate \( f(f(x)) = f(x + a) \).

Step 3: Substitute and Simplify

Substitute \( x + a \) back into \( f(x) \): \( f(x + a) = (x + a) + a \). Simplify this to get \( x + 2a \).

Step 4: Verify the Solution

We found that \( f(f(x)) = x + 2a \) by applying the function twice. We verified the steps to ensure no mistakes were made.

Key Concepts

Function Evaluation

Function evaluation is the process of calculating the output of a function for a given input. It's like following a recipe to get a desired dish by inserting specific ingredients. In our exercise, the function is given as \( f(x) = x + a \). This means that for any input \( x \), we simply add \( a \) to it to find the output.

• If \( x \) is 2 and \( a \) is 3, then \( f(2) = 2 + 3 = 5 \).
• No matter what value \( x \) or \( a \) takes, evaluating \( f(x) \) involves performing this basic addition.

To determine \( f(x) \), substitute the given \( x \) into the equation and solve. This task of evaluating functions is fundamental in mathematics and builds the foundation for more complex operations like function composition.

Function Composition

Function composition refers to applying one function to the results of another. It's like putting one process inside another. When we compose two functions, we apply one function and then apply another function to the result. This can be visualized as chaining processes together.In our exercise, since \( f(x) = x + a \), function composition means finding \( f(f(x)) \). This involves applying \( f \) to the result of \( f(x) \):

• First, find \( f(x) = x + a \).
• Second, apply \( f \) again to this result to find \( f(x + a) \).

In simpler terms, composing a function with itself means taking the original formula and replacing \( x \) with the output of function applied to \( x \).Function composition is a crucial concept in algebra, allowing us to construct new functions by combining existing ones. This idea is widely used for simplifying or modifying mathematical expressions.

Algebraic Manipulation

Algebraic manipulation involves using mathematical operations and transformations to simplify or rearrange expressions. Consider it like reorganizing legos to create a different structure. In our exercise, we practiced algebraic manipulation while handling the composition \( f(f(x)) = f(x + a) = (x + a) + a \).Here are some steps involved in this manipulation:

• Start by substituting the result of the original function \( f(x) = x + a \) into itself to get \( f(x + a) \).
• Then, recognize that \( f(x + a) \) means adding another \( a \), resulting in \( x + a + a \).
• Simplify the expression by combining like terms, which gives \( x + 2a \).

Without algebraic manipulation, it's challenging to solve or simplify equations. It's a powerful tool for reinterpreting complex equations into more manageable forms, which is extremely helpful in making sense of mathematical relationships.