Finding (f-g)(x) And (f-g)(5) With F(x) = X² - 4x And G(x) = 4 - X A Step-by-Step Guide

Jul 14, 2025 by ADMIN 88 views

Understanding the Difference of Functions: A Comprehensive Guide to (f-g)(x) and (f-g)(5) Given f(x) = x² - 4x and g(x) = 4 - x

In the realm of mathematics, functions serve as fundamental building blocks for modeling relationships between variables. Operations on functions, such as subtraction, allow us to analyze how these relationships interact. This article delves into the process of finding the difference of two functions, denoted as (f-g)(x), and evaluating this difference at a specific point, (f-g)(5), given the functions f(x) = x² - 4x and g(x) = 4 - x. This comprehensive guide aims to provide a clear understanding of the concepts involved and a step-by-step approach to solving such problems.

Defining Function Subtraction: (f-g)(x)

The operation (f-g)(x) represents the subtraction of the function g(x) from the function f(x). In simpler terms, for any input value 'x', we subtract the output of g(x) from the output of f(x). Mathematically, this is expressed as:

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

To find (f-g)(x), we need to substitute the expressions for f(x) and g(x) into this equation and simplify the resulting expression. This process involves algebraic manipulation, such as combining like terms, to arrive at a simplified expression for the new function (f-g)(x). Let's apply this to our given functions.

Given our functions f(x) = x² - 4x and g(x) = 4 - x, we can find (f-g)(x) by substituting these expressions into the formula:

(f-g)(x) = (x² - 4x) - (4 - x)

The next step is to distribute the negative sign to the terms within the parentheses of g(x):

(f-g)(x) = x² - 4x - 4 + x

Now, we combine like terms, which in this case are the terms involving 'x':

(f-g)(x) = x² - 4x + x - 4

(f-g)(x) = x² - 3x - 4

Therefore, the difference of the functions, (f-g)(x), is the quadratic function x² - 3x - 4. This new function represents the vertical difference between the graphs of f(x) and g(x) at each point 'x'. Understanding this resulting function allows us to analyze the relationship between the original functions and predict their behavior.

Evaluating the Difference: Finding (f-g)(5)

Once we have determined the expression for (f-g)(x), we can evaluate it at a specific value of 'x'. This means substituting that value for 'x' in the expression and simplifying to obtain a numerical result. In our case, we want to find (f-g)(5), which means we need to substitute x = 5 into the expression we found for (f-g)(x).

We found that (f-g)(x) = x² - 3x - 4. To find (f-g)(5), we substitute x = 5 into this equation:

(f-g)(5) = (5)² - 3(5) - 4

Now, we perform the arithmetic operations:

(f-g)(5) = 25 - 15 - 4

(f-g)(5) = 10 - 4

(f-g)(5) = 6

Thus, the value of the difference of the functions at x = 5, (f-g)(5), is 6. This means that at x = 5, the difference between the outputs of f(x) and g(x) is 6. This value represents a specific point on the graph of the function (f-g)(x) and provides insight into the relationship between f(x) and g(x) at that particular point.

Step-by-Step Solution

Let's recap the entire process with a step-by-step solution:

Write down the functions:

f(x) = x² - 4x

g(x) = 4 - x
Write the expression for (f-g)(x):

(f-g)(x) = f(x) - g(x)
Substitute the expressions for f(x) and g(x):

(f-g)(x) = (x² - 4x) - (4 - x)
Distribute the negative sign:

(f-g)(x) = x² - 4x - 4 + x
Combine like terms:

(f-g)(x) = x² - 3x - 4
Substitute x = 5 into (f-g)(x):

(f-g)(5) = (5)² - 3(5) - 4
Simplify:

(f-g)(5) = 25 - 15 - 4

(f-g)(5) = 6

This step-by-step approach provides a clear and organized method for solving problems involving the subtraction of functions and their evaluation at specific points. By following these steps, you can confidently tackle similar problems.

Visualizing Function Subtraction

Visualizing function subtraction can provide a deeper understanding of the concept. Consider the graphs of f(x) = x² - 4x and g(x) = 4 - x. The graph of (f-g)(x) represents the vertical distance between the graphs of f(x) and g(x) at each point 'x'.

At a given 'x' value, if f(x) is greater than g(x), then (f-g)(x) will be positive. If g(x) is greater than f(x), then (f-g)(x) will be negative. And if f(x) and g(x) have the same value, then (f-g)(x) will be zero. This visual representation helps to connect the algebraic manipulation with the geometric interpretation of function subtraction.

In our example, (f-g)(x) = x² - 3x - 4 is a parabola. The value (f-g)(5) = 6 represents the y-coordinate of the point on this parabola where x = 5. By plotting the graphs of f(x), g(x), and (f-g)(x), you can visually confirm that the vertical distance between f(x) and g(x) at x = 5 is indeed 6.

Common Mistakes to Avoid

When working with function subtraction, there are a few common mistakes that students often make. Being aware of these pitfalls can help you avoid them and ensure accurate solutions:

Incorrectly distributing the negative sign: A crucial step in finding (f-g)(x) is distributing the negative sign to all terms within the parentheses of g(x). Forgetting to do so or distributing it incorrectly will lead to an incorrect result. For example, in our problem, it's essential to distribute the negative sign in -(4 - x) to get -4 + x.
Combining unlike terms: Only like terms can be combined. For instance, terms with x² cannot be combined with terms with x or constant terms. Ensure you are only combining terms with the same variable and exponent.
Substituting incorrectly: When evaluating (f-g)(x) at a specific value, ensure you are substituting the value correctly for every instance of 'x' in the expression. A simple mistake in substitution can lead to a wrong answer.
Arithmetic errors: Be careful when performing arithmetic operations, especially when dealing with negative numbers. Double-check your calculations to avoid errors in simplification and evaluation.

By being mindful of these common mistakes and practicing the steps carefully, you can improve your accuracy in solving function subtraction problems.

Applications of Function Subtraction

Function subtraction is not just a mathematical exercise; it has various practical applications in different fields. One common application is in cost-benefit analysis.

For example, let's say f(x) represents the revenue generated by selling 'x' units of a product, and g(x) represents the cost of producing 'x' units. Then (f-g)(x) represents the profit made from selling 'x' units. By analyzing this difference function, businesses can determine the number of units they need to sell to break even or maximize their profit.

Another application is in physics. If f(t) represents the position of an object at time 't', and g(t) represents the position of another object at the same time, then (f-g)(t) represents the relative position of the two objects. This can be used to analyze the motion of objects relative to each other.

Function subtraction also finds applications in areas like signal processing, image analysis, and economics. Understanding the concept of function subtraction provides a powerful tool for modeling and analyzing real-world situations.

Practice Problems

To solidify your understanding of function subtraction, here are a few practice problems:

Given f(x) = 2x² + 3x - 1 and g(x) = x - 5, find (f-g)(x) and (f-g)(2).
Given f(x) = √x and g(x) = x - 2, find (f-g)(x) and (f-g)(4).
Given f(x) = |x| and g(x) = x + 1, find (f-g)(x) and (f-g)(-1).

Working through these problems will help you reinforce the steps involved in function subtraction and improve your problem-solving skills. Remember to follow the step-by-step approach outlined earlier and be mindful of common mistakes.

Conclusion

In this article, we have explored the concept of function subtraction, specifically finding (f-g)(x) and evaluating it at a point, (f-g)(5). We began by defining function subtraction and outlining the algebraic steps involved in finding the difference of two functions. We then applied these steps to the specific functions f(x) = x² - 4x and g(x) = 4 - x, finding that (f-g)(x) = x² - 3x - 4 and (f-g)(5) = 6.

We also discussed the importance of visualizing function subtraction and how the graph of (f-g)(x) represents the vertical distance between the graphs of f(x) and g(x). We highlighted common mistakes to avoid when performing function subtraction and explored real-world applications of this concept.

By understanding the principles and techniques presented in this guide, you will be well-equipped to solve problems involving function subtraction and apply this knowledge in various mathematical and practical contexts. Remember to practice regularly and seek help when needed to further enhance your understanding and skills in this area.