Calculating (f-g)(x): A Simple Guide With Examples

Nov 30, 2025 by Alex Johnson 51 views

Have you ever wondered how to combine two functions in mathematics? One common operation is finding the difference between two functions, denoted as (f-g)(x). This guide will walk you through the process step-by-step, using a specific example to make it crystal clear. Let's dive in!

Understanding Function Notation

Before we jump into the calculation, it's important to understand the notation. In mathematics, a function is like a machine that takes an input (x) and produces an output. We often write functions as f(x), where 'f' is the name of the function and 'x' is the input variable. The expression f(x) represents the output of the function f when the input is x. Similarly, g(x) represents another function with the input x.

Key Concepts of Function Notation

Understanding function notation is crucial for grasping the concept of (f-g)(x). Here's a breakdown of the key concepts:

Function as a Machine: Think of a function as a machine. You put something in (the input), and the machine does something to it and gives you something out (the output).
f(x): This notation means "the value of the function f at x." It's the output you get from the function f when you input x.
Input Variable: The 'x' in f(x) is the input variable. It's the value that you're feeding into the function.
Output: The result of applying the function to the input is the output. This is what f(x) equals.
Different Functions: You can have many different functions, each with its own rule for transforming the input. We use different letters (like f, g, h) to represent different functions.

For example, if we have f(x) = 2x + 1, this means that the function f takes an input x, multiplies it by 2, and then adds 1. So, if we input x = 3, we get f(3) = 2(3) + 1 = 7. The output is 7.

Similarly, g(x) could be another function, like g(x) = x^2. This function takes an input x and squares it. If we input x = 4, we get g(4) = 4^2 = 16. The output is 16.

Understanding this basic notation is essential for performing operations on functions, such as finding (f-g)(x). It allows us to clearly express the relationship between inputs and outputs and to manipulate functions algebraically.

What is (f-g)(x)?

The expression (f-g)(x) represents a new function formed by subtracting the function g(x) from the function f(x). In simpler terms, you take the output of f(x) and subtract the output of g(x) for the same input x. Mathematically, it's defined as:

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

This means that to find (f-g)(x), you need to:

Identify the expressions for f(x) and g(x).
Subtract the expression for g(x) from the expression for f(x).
Simplify the resulting expression.

Let's illustrate this with the example provided: f(x) = 3x and g(x) = -3x + 5.

Why is (f-g)(x) Important?

The operation (f-g)(x) is a fundamental concept in mathematics with numerous applications across various fields. Understanding how to subtract functions provides valuable insights into the relationships between them and allows for more complex analysis and modeling. Here are some key reasons why (f-g)(x) is important:

Analyzing Differences: (f-g)(x) allows us to analyze the difference between two functions. This is particularly useful in scenarios where you want to compare the outputs of two different processes or models for the same input. For instance, in economics, f(x) might represent the revenue function and g(x) might represent the cost function. Then, (f-g)(x) would represent the profit function, showing the difference between revenue and cost.
Modeling Change: In many real-world situations, we need to model how quantities change over time or with respect to some other variable. By subtracting functions, we can isolate specific components of change. For example, if f(x) represents the total sales and g(x) represents the marketing expenses, (f-g)(x) could represent the sales without the influence of marketing, helping to evaluate the effectiveness of the marketing strategy.
Simplifying Complex Functions: Sometimes, dealing with individual functions can be cumbersome, especially when they are part of a larger system. Subtracting functions can help simplify complex expressions and make them more manageable. This is particularly useful in calculus, where understanding the derivatives and integrals of simpler functions can make complex problems easier to solve.
Solving Equations: The concept of subtracting functions is also crucial in solving equations. By setting (f-g)(x) equal to zero, we can find the values of x where the functions f(x) and g(x) are equal. This is a common technique in algebra and calculus for finding intersection points or equilibrium points.
Optimization Problems: In optimization problems, where the goal is to maximize or minimize a certain quantity, (f-g)(x) can be used to define objective functions. For example, if f(x) represents the benefits and g(x) represents the costs, then maximizing (f-g)(x) would lead to the most profitable or efficient outcome.

In summary, understanding and calculating (f-g)(x) is essential for a wide range of mathematical applications. It provides a powerful tool for analyzing differences, modeling change, simplifying expressions, solving equations, and optimizing outcomes, making it a cornerstone of mathematical analysis and problem-solving.

Step-by-Step Calculation of (f-g)(x)

Now, let's apply the concept to the specific functions given:

f(x) = 3x g(x) = -3x + 5

To find (f-g)(x), we substitute these expressions into the formula:

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

(f-g)(x) = (3x) - (-3x + 5)

Notice the parentheses around the expression for g(x). This is crucial because we need to subtract the entire expression, not just the first term. Now, let's distribute the negative sign:

(f-g)(x) = 3x + 3x - 5

Combine like terms:

(f-g)(x) = 6x - 5

So, (f-g)(x) = 6x - 5. This is a linear function in simplest form.

Detailed Steps with Explanations

To ensure a clear understanding of how (f-g)(x) is calculated, let's break down the process into detailed steps, providing explanations for each. This step-by-step approach is especially helpful for those new to the concept or for anyone looking to reinforce their understanding.

Step 1: Identify the Functions

The first step in calculating (f-g)(x) is to clearly identify the functions f(x) and g(x). This involves writing down the expressions for each function. In our example, we have:

f(x) = 3x g(x) = -3x + 5

Identifying the functions correctly is the foundation of the entire process. Without this step, you won't know what expressions to work with in the subsequent steps.

Step 2: Write the Formula

Next, write down the general formula for (f-g)(x), which is:

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

This formula serves as a roadmap for the calculation. It reminds you that you need to subtract the entire expression of g(x) from f(x).

Step 3: Substitute the Function Expressions

Now, substitute the expressions for f(x) and g(x) into the formula. Be sure to use parentheses around the expression for g(x) to ensure that the entire expression is subtracted. This is a critical step to avoid errors with the signs.

(f-g)(x) = (3x) - (-3x + 5)

Step 4: Distribute the Negative Sign

The next step is to distribute the negative sign across the terms inside the parentheses. This means multiplying each term in the expression for g(x) by -1. This is a common point of error, so be extra careful here.

(f-g)(x) = 3x + 3x - 5

Notice how -(-3x) becomes +3x and -(+5) becomes -5. Distributing the negative sign correctly is essential for getting the right answer.

Step 5: Combine Like Terms

The final step is to combine like terms. Like terms are terms that have the same variable raised to the same power. In our expression, 3x and 3x are like terms. Combine them by adding their coefficients:

(f-g)(x) = (3x + 3x) - 5 (f-g)(x) = 6x - 5

The simplified expression 6x - 5 is the result of subtracting g(x) from f(x).

Step 6: Final Result

The final result is the simplified expression for (f-g)(x). In this case, we have:

(f-g)(x) = 6x - 5

This is the answer to the problem. The function (f-g)(x) is a linear function with a slope of 6 and a y-intercept of -5.

By following these detailed steps, you can confidently calculate (f-g)(x) for any given functions f(x) and g(x). Remember to pay close attention to the signs and to combine like terms correctly. This methodical approach will help you avoid common mistakes and ensure accurate results.

Presenting the Answer

The final answer, (f-g)(x) = 6x - 5, is a polynomial in simplest form. It represents a linear function with a slope of 6 and a y-intercept of -5. This means that for every increase of 1 in x, the value of (f-g)(x) increases by 6, and the line crosses the y-axis at the point (0, -5).

Additional Examples

Let's explore a few more examples to solidify your understanding of calculating (f-g)(x). These examples will cover different types of functions, including quadratic and constant functions, to illustrate the versatility of the process.

Example 1: Quadratic and Linear Functions

Suppose we have the functions:

f(x) = x^2 + 3x - 2 g(x) = 2x + 1

To find (f-g)(x), we follow the same steps as before:

Write the formula: (f-g)(x) = f(x) - g(x)
Substitute the expressions: (f-g)(x) = (x^2 + 3x - 2) - (2x + 1)
Distribute the negative sign: (f-g)(x) = x^2 + 3x - 2 - 2x - 1
Combine like terms: (f-g)(x) = x^2 + (3x - 2x) + (-2 - 1)
Simplify: (f-g)(x) = x^2 + x - 3

So, in this case, (f-g)(x) is a quadratic function, x^2 + x - 3.

Example 2: Constant and Linear Functions

Let's consider the functions:

f(x) = 7 g(x) = -x + 4

Here, f(x) is a constant function, meaning its output is always 7, regardless of the input x. To find (f-g)(x):

Write the formula: (f-g)(x) = f(x) - g(x)
Substitute the expressions: (f-g)(x) = (7) - (-x + 4)
Distribute the negative sign: (f-g)(x) = 7 + x - 4
Combine like terms: (f-g)(x) = x + (7 - 4)
Simplify: (f-g)(x) = x + 3

Therefore, (f-g)(x) = x + 3, which is a linear function.

Example 3: Two Quadratic Functions

Now, let's look at two quadratic functions:

f(x) = 2x^2 - x + 5 g(x) = x^2 + 4x - 1

To calculate (f-g)(x):

Write the formula: (f-g)(x) = f(x) - g(x)
Substitute the expressions: (f-g)(x) = (2x^2 - x + 5) - (x^2 + 4x - 1)
Distribute the negative sign: (f-g)(x) = 2x^2 - x + 5 - x^2 - 4x + 1
Combine like terms: (f-g)(x) = (2x^2 - x^2) + (-x - 4x) + (5 + 1)
Simplify: (f-g)(x) = x^2 - 5x + 6

In this example, (f-g)(x) = x^2 - 5x + 6, which is another quadratic function.

These additional examples demonstrate how to calculate (f-g)(x) for a variety of functions. The key is to follow the steps carefully: identify the functions, write the formula, substitute the expressions, distribute the negative sign, and combine like terms. With practice, this process will become second nature.

Understanding how to subtract functions is a fundamental skill in mathematics, and these examples provide a solid foundation for more advanced topics.

Conclusion

Calculating (f-g)(x) is a straightforward process once you understand the basic principles. It involves subtracting the expression for g(x) from the expression for f(x) and simplifying the result. By following the steps outlined in this guide, you can confidently calculate (f-g)(x) for any given functions. Remember to pay close attention to the signs and combine like terms carefully. This skill is fundamental in algebra and calculus, and mastering it will open doors to more advanced mathematical concepts.

For further exploration and practice, you might find helpful resources on websites like Khan Academy's Function Operations. This will provide you with additional examples and exercises to enhance your understanding.