Problem 35 - Calculus with Concepts in Calculus

Question

Find \(g\) if \(f(x)=|x|\) and \((f+g)(x)=|x|-|x-2|\).

Step-by-Step Solution

Answer

\(g(x) = -|x-2|\).

Step 1: Identify what is given and what to find

The problem gives us a function \(f(x) = |x|\) and a composite function \((f+g)(x) = |x| - |x-2|\). We are asked to find the function \(g(x)\).

Step 2: Express the composite function using given functions

According to the problem, \((f+g)(x) = f(x) + g(x)\). This implies that \(|x| - |x-2| = |x| + g(x)\). We need to rearrange this to express \(g(x)\) in terms of \(f(x)\).

Step 3: Solve for \(g(x)\)

From the equation \(|x| + g(x) = |x| - |x-2|\), subtract \(|x|\) from both sides to isolate \(g(x)\): \(g(x) = -|x-2|\).

Step 4: Verify the solution

The solution \(g(x) = -|x-2|\) should satisfy the original equation \((f+g)(x) = |x| - |x-2|\). Substitute \(g(x)\) back into the equation: \(|x| + (-|x-2|) = |x| - |x-2|\). The equation holds true, confirming our solution is correct.

Key Concepts

Understanding Absolute Value Functions

An absolute value function is defined as the distance of a number from zero on the number line. This implies it is always non-negative. The absolute value of a number, say \(x\), is written as \(|x|\).
It can also be defined piecewise as:

• \(|x| = x\) if \(x \geq 0\)
• \(|x| = -x\) if \(x < 0\)

The absolute value function is important in calculus because it behaves differently on either side of zero. When dealing with absolute values, you need to consider different cases for different domains. For example, \(|x-2|\) represents the distance from \(x\) to 2, which can change depending on whether \(x\) is less than or greater than 2.
This specific behavior makes absolute value functions appear often in piecewise functions, which means they have different expressions based on the input value's range.

Introduction to Composite Functions

In calculus, a composite function is a function formed when one function is applied to the result of another function. If \(f\) and \(g\) are functions, then the composite function \((f+g)(x)\) means you apply both \(f\) and \(g\) to the input \(x\) and add the results.
This can be represented as:\[(f+g)(x) = f(x) + g(x)\]Composite functions are useful for understanding how functions interact and change their shape or behavior when combined. By breaking them down into their components, we can study their individual effects or changes.
In our exercise, the task was to break down the equation \((f+g)(x) = |x| - |x-2|\) to find the individual function \(g(x)\) that, together with \(f(x)\), produces the given composite function.

Function Operations Explained

Function operations allow us to manipulate and combine functions to create new ones. The most common operations include addition, subtraction, multiplication, and division of functions. When we talk about these operations, we apply them over the entire range of input values.
In the given problem, we focus on addition, where the functions are combined by adding their outputs:

• \((f+g)(x) = f(x) + g(x)\)
• In our case, \(|x| + g(x) = |x| - |x-2|\)

By rearranging this equation, we isolate \(g(x)\) as \(-|x-2|\). This subtraction within the composite function demonstrates how function operations can allow us to find unknown functions by analyzing and rearranging the given expressions. Function operations are fundamental in calculus because they help us deconstruct complex problems into simpler parts.
By understanding and mastering these operations, you'll be able to solve a wide range of calculus problems involving different types of functions.