Function Subtraction Explained Finding (f-g)(x) Step By Step
In the realm of mathematics, particularly in algebra, functions play a crucial role. Functions are mathematical relationships that map inputs to outputs. They can be combined and manipulated in various ways, including addition, subtraction, multiplication, and division. Understanding these operations is fundamental for solving complex mathematical problems. In this article, we will delve into the concept of function subtraction and how to find (f-g)(x) given two functions, f(x) and g(x).
Defining Function Subtraction
Function subtraction, denoted as (f-g)(x), involves subtracting the function g(x) from the function f(x). This operation is performed by subtracting the expressions that define the functions. Mathematically, it can be expressed as:
(f-g)(x) = f(x) - g(x)
This means that for any given value of x, the value of (f-g)(x) is obtained by subtracting the value of g(x) at that x from the value of f(x) at the same x.
Step-by-Step Solution for Finding (f-g)(x)
To illustrate the process of function subtraction, let's consider the specific example provided: f(x) = 3x^2 - 2 and g(x) = 2x + 4. Our goal is to find (f-g)(x).
1. Write down the expressions for f(x) and g(x).
We are given:
- f(x) = 3x^2 - 2
- g(x) = 2x + 4
2. Substitute the expressions into the formula (f-g)(x) = f(x) - g(x).
Substituting the given expressions, we get:
(f-g)(x) = (3x^2 - 2) - (2x + 4)
3. Distribute the negative sign to the terms in g(x).
Distributing the negative sign in front of the parenthesis, we have:
(f-g)(x) = 3x^2 - 2 - 2x - 4
4. Combine like terms.
Now, we combine the like terms (terms with the same power of x):
(f-g)(x) = 3x^2 - 2x - 2 - 4
(f-g)(x) = 3x^2 - 2x - 6
Therefore, (f-g)(x) = 3x^2 - 2x - 6.
Analyzing the Result
The result, (f-g)(x) = 3x^2 - 2x - 6, is a quadratic function. This is because the highest power of x in the expression is 2. The subtraction of two functions can result in a new function with a different degree and characteristics than the original functions. In this case, we started with a quadratic function (f(x)) and a linear function (g(x)), and their subtraction resulted in another quadratic function.
Understanding the properties of the resulting function, such as its degree, leading coefficient, and intercepts, can provide further insights into the relationship between the original functions.
Common Mistakes to Avoid
When performing function subtraction, it's crucial to avoid common mistakes that can lead to incorrect results. Here are some common pitfalls:
1. Forgetting to distribute the negative sign.
One of the most frequent errors is failing to distribute the negative sign to all terms in g(x). Remember that subtracting a function is equivalent to adding the negative of that function. Therefore, the negative sign must be applied to every term within the parentheses of g(x).
For example, in the expression (3x^2 - 2) - (2x + 4), it's essential to distribute the negative sign to both 2x and 4, resulting in -2x - 4. Failing to do so would lead to an incorrect result.
2. Incorrectly combining like terms.
Another common mistake is combining terms that are not like terms. Like terms are terms that have the same variable raised to the same power. For instance, 3x^2 and -2x are not like terms because they have different powers of x. Similarly, -2 and -4 are like terms because they are both constants.
When combining like terms, ensure that you only add or subtract the coefficients of the terms with the same variable and power. For example, 3x^2 - 2x - 2 - 4 should be simplified to 3x^2 - 2x - 6, where the constants -2 and -4 are combined.
3. Misunderstanding the order of operations.
The order of operations (PEMDAS/BODMAS) is crucial in mathematical calculations. It dictates the sequence in which operations should be performed: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).
In function subtraction, it's essential to follow the order of operations correctly. First, simplify any expressions within parentheses, then perform the subtraction operation, and finally combine like terms.
4. Not paying attention to signs.
Sign errors are common in algebra, especially when dealing with negative numbers. Pay close attention to the signs of each term when distributing the negative sign and combining like terms. A simple sign error can significantly alter the final result.
For example, when distributing the negative sign in (3x^2 - 2) - (2x + 4), ensure that you correctly change the signs of both 2x and 4. The expression should become 3x^2 - 2 - 2x - 4, not 3x^2 - 2 - 2x + 4.
5. Skipping steps.
Skipping steps in algebraic manipulations can increase the likelihood of making errors. It's best to write out each step clearly and systematically to avoid mistakes. This is especially important when dealing with more complex expressions.
By avoiding these common mistakes, you can improve your accuracy and confidence in performing function subtraction.
Practice Problems
To solidify your understanding of function subtraction, try solving these practice problems:
- If f(x) = 5x^2 + 3x - 1 and g(x) = x^2 - 2x + 3, find (f-g)(x).
- Given f(x) = 4x - 7 and g(x) = -2x + 5, determine (f-g)(x).
- Let f(x) = x^3 + 2x^2 - x + 4 and g(x) = 3x^2 + 2x - 1, find (f-g)(x).
By working through these problems, you can reinforce your skills and gain a deeper understanding of function subtraction.
Conclusion
Function subtraction is a fundamental operation in algebra that allows us to combine and manipulate functions. By understanding the process of subtracting functions and avoiding common mistakes, you can confidently solve problems involving function subtraction. This skill is essential for further studies in mathematics and related fields.
Remember, the key to mastering function subtraction is practice. Work through various examples, pay attention to detail, and don't hesitate to seek help when needed. With consistent effort, you can develop a strong understanding of function subtraction and its applications.
This article has provided a comprehensive guide to function subtraction, covering the definition, step-by-step solution, common mistakes to avoid, and practice problems. By applying the concepts and techniques discussed, you can enhance your mathematical skills and excel in algebra.
Final Answer
The final answer is (D) .
