# (f-g)(x): How to Solve It

## (f-g)(x): How to Solve It

## Introduction

When studying mathematics, especially algebra and calculus, one often encounters functions and their operations. One such operation is the subtraction of two functions, commonly denoted as $(f-g)(x)$. Understanding how to solve $(f-g)(x)$ is fundamental for students and professionals dealing with mathematical models. This article aims to provide a comprehensive guide on solving $(f-g)(x)$, with clear explanations and examples.

## What is $(f-g)(x)$?

Before diving into solving $(f-g)(x)$, let's clarify what it represents. If $f(x)$ and $g(x)$ are two functions, then $(f-g)(x)$ denotes the function obtained by subtracting $g(x)$ from $f(x)$. Mathematically, it can be expressed as:

$(f-g)(x) = f(x) - g(x)$

This operation results in a new function that represents the difference between the two original functions at any given value of $x$.

## Steps to Solve $(f-g)(x)$

### 1. Understand the Functions

The first step is to clearly understand the functions $f(x)$ and $g(x)$. These functions can be any type of mathematical expressions, including polynomials, trigonometric functions, exponential functions, etc.

**Example:**
Let's consider the following functions:
$f(x) = 3x^2 + 2x - 5$
$g(x) = x^2 - 4x + 1$

### 2. Identify the Domain

The domain of $(f-g)(x)$ is the set of all $x$ values for which both $f(x)$ and $g(x)$ are defined. If there are any restrictions on the domain of either function, these must be taken into account.

**Example:**
For the functions $f(x) = 3x^2 + 2x - 5$ and $g(x) = x^2 - 4x + 1$, both are polynomials, which are defined for all real numbers. Hence, the domain is all real numbers ($\mathbb{R}$).

### 3. Subtract the Functions

Next, subtract $g(x)$ from $f(x)$ to find $(f-g)(x)$.

**Example:**
$(f-g)(x) = (3x^2 + 2x - 5) - (x^2 - 4x + 1)$

### 4. Simplify the Expression

Combine like terms to simplify the expression.

**Example:**

So, $(f-g)(x) = 2x^2 + 6x - 6$.

### 5. Verify the Result

Finally, verify your result by checking specific values of $x$ in both the original and the resultant functions.

**Example:**
Let's verify by substituting $x = 1$:
$f(1) = 3(1)^2 + 2(1) - 5 = 3 + 2 - 5 = 0$
$g(1) = (1)^2 - 4(1) + 1 = 1 - 4 + 1 = -2$
$(f-g)(1) = f(1) - g(1) = 0 - (-2) = 2$
$2(1)^2 + 6(1) - 6 = 2 + 6 - 6 = 2$

Both approaches give the same result, verifying the correctness of our solution.

## Practical Examples

### Example 1: Subtracting Linear Functions

Given $f(x) = 5x + 3$ and $g(x) = 2x - 4$: $(f-g)(x) = (5x + 3) - (2x - 4)$ $(f-g)(x) = 5x + 3 - 2x + 4$ $(f-g)(x) = 3x + 7$

### Example 2: Subtracting Trigonometric Functions

Given $f(x) = \sin(x)$ and $g(x) = \cos(x)$: $(f-g)(x) = \sin(x) - \cos(x)$

### Example 3: Subtracting Exponential Functions

Given $f(x) = e^x$ and $g(x) = 2e^x$: $(f-g)(x) = e^x - 2e^x$ $(f-g)(x) = -e^x$

## Applications of $(f-g)(x)$

### 1. Engineering

In engineering, particularly in control systems, the difference between functions can represent the error between desired and actual system responses.

### 2. Economics

Economists use the difference between supply and demand functions to determine equilibrium points and analyze market behavior.

### 3. Physics

Physicists use function subtraction to analyze wave interference patterns, where the resultant wave is the difference of two overlapping waves.

## Conclusion

Solving $(f-g)(x)$ is a fundamental skill in mathematics, with wide-ranging applications across various fields. By understanding the functions, identifying the domain, subtracting the functions, simplifying the expression, and verifying the result, you can effectively solve these types of problems.

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