(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.