Finding (f-g)(x) Step-by-Step Solution

In the realm of mathematics, functions serve as fundamental building blocks for modeling relationships between variables. Operations on functions, such as addition, subtraction, multiplication, and division, allow us to combine and manipulate functions to create new ones. In this comprehensive guide, we will delve into the process of finding the difference of two functions, specifically focusing on the scenario where $f(x) = -5^x - 4$ and $g(x) = -3x - 2$. Understanding how to perform this operation is crucial for various mathematical applications, including calculus, algebra, and data analysis.

Understanding Function Operations

Before we dive into the specifics of finding $(f-g)(x)$, let's first establish a solid understanding of function operations in general. Just like we can perform arithmetic operations on numbers, we can also perform operations on functions. These operations include:

• Addition: $(f + g)(x) = f(x) + g(x)$
• Subtraction: $(f - g)(x) = f(x) - g(x)$
• Multiplication: $(f imes g)(x) = f(x) imes g(x)$
• Division: $(f / g)(x) = f(x) / g(x)$, where $g(x) e 0$

The notation $(f - g)(x)$ represents the difference between the functions $f(x)$ and $g(x)$. This means we subtract the expression for $g(x)$ from the expression for $f(x)$. It's crucial to remember the order of operations and pay close attention to signs when performing this subtraction.

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

Now that we have a firm grasp of function operations, let's apply this knowledge to find $(f-g)(x)$ for the given functions $f(x) = -5^x - 4$ and $g(x) = -3x - 2$. We will meticulously walk through each step to ensure clarity and understanding.

1. Write the Definition of (f-g)(x)

The first step is to write down the definition of $(f-g)(x)$, which is:

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

This definition serves as our roadmap for the subsequent steps. It clearly states that we need to subtract the expression for $g(x)$ from the expression for $f(x)$.

2. Substitute the Expressions for f(x) and g(x)

Next, we substitute the given expressions for $f(x)$ and $g(x)$ into the equation:

$(f - g)(x) = (-5^x - 4) - (-3x - 2)$

This step replaces the function notations with their corresponding algebraic expressions, setting the stage for simplification.

3. Distribute the Negative Sign

Here's where careful attention to signs is paramount. We need to distribute the negative sign in front of the parentheses containing $g(x)$. Remember that subtracting a negative is the same as adding a positive. So, we have:

$(f - g)(x) = -5^x - 4 + 3x + 2$

Distributing the negative sign correctly is crucial for arriving at the accurate result. A common mistake is to forget to distribute the negative sign to both terms within the parentheses.

4. Combine Like Terms

The final step is to combine any like terms. In this case, we can combine the constant terms -4 and +2:

$(f - g)(x) = -5^x + 3x - 2$

This step simplifies the expression by grouping similar terms together, resulting in the final form of $(f-g)(x)$.

Therefore, $(f-g)(x) = -5^x + 3x - 2$.

Common Mistakes to Avoid

When working with function operations, it's easy to make mistakes if you're not careful. Here are some common pitfalls to watch out for:

• Forgetting to Distribute the Negative Sign: This is a frequent error when subtracting functions. Make sure to distribute the negative sign to every term within the parentheses.
• Incorrect Order of Operations: Follow the order of operations (PEMDAS/BODMAS) when simplifying expressions. Exponents should be evaluated before addition and subtraction.
• Combining Unlike Terms: Only combine terms that have the same variable and exponent. For instance, you cannot combine $-5^x$ and $3x$ because they are not like terms.
• Sign Errors: Pay close attention to signs throughout the calculation. A small sign error can lead to a completely different answer.

Practice Problems

To solidify your understanding of function subtraction, try working through these practice problems:

1. If $f(x) = 2x^2 + 3x - 1$ and $g(x) = x^2 - 2x + 4$, find $(f - g)(x)$.
2. If f(x) = rac{1}{x} and g(x) = rac{x}{x+1}, find $(f - g)(x)$.
3. If $f(x) = ext{sqrt}(x+2)$ and $g(x) = x - 3$, find $(f - g)(x)$.

Working through these problems will help you develop confidence and proficiency in subtracting functions.

Applications of Function Subtraction

Function subtraction isn't just an abstract mathematical concept; it has practical applications in various fields. Here are a few examples:

• Profit Calculation: In business, profit is calculated as revenue minus cost. If we have functions representing revenue and cost, we can use function subtraction to find the profit function.
• Rate of Change: In calculus, the difference of functions can be used to find the rate of change between two quantities. For example, we might subtract a function representing distance traveled from a function representing time to find the average speed.
• Data Analysis: In statistics, function subtraction can be used to compare different datasets or models. For example, we might subtract one regression model from another to see which one fits the data better.

Conclusion

Finding the difference of functions is a fundamental operation in mathematics with wide-ranging applications. By following a step-by-step approach, paying attention to detail, and practicing regularly, you can master this skill and confidently apply it to solve various problems. Remember to distribute the negative sign carefully, combine like terms correctly, and avoid common mistakes. With practice, you'll become proficient in subtracting functions and unlock a powerful tool for mathematical analysis.