Find (f-g)(x) Given F(x) And G(x)

Nov 11, 2025 by ADMIN 34 views

Hey guys! Today, we're diving into a super fun problem from the world of functions. Specifically, we're going to figure out how to find $(f-g)(x)$ when we're given two functions, $f(x)$ and $g(x)$ . It might sound a bit intimidating at first, but trust me, it's easier than it looks! So, let's jump right in.

Understanding the Basics

Before we get started, let's quickly recap what functions are and how they work. A function is like a machine that takes an input (usually represented by 'x') and spits out an output based on a specific rule. For example, if we have $f(x) = 2x + 3$ , it means that whatever number we plug in for 'x', we multiply it by 2 and then add 3 to get the result.

Now, what does $(f-g)(x)$ even mean? Well, it's simply a shorthand way of saying "take the function $f(x)$ , subtract the function $g(x)$ from it, and see what you get." In other words, $(f-g)(x) = f(x) - g(x)$ . This is a fundamental operation in dealing with functions, and mastering it opens doors to more complex mathematical concepts. The beauty of this operation lies in its simplicity; it's a straightforward application of algebraic principles. The expression $(f-g)(x)$ represents a new function formed by subtracting the values of function $g$ from function $f$ at each point $x$ . This concept is widely used in various fields of mathematics, including calculus, where understanding the behavior of combined functions is crucial. By manipulating functions in this way, we can model real-world phenomena, such as the difference in production costs between two companies or the net change in population size over time. Therefore, a solid grasp of this operation is essential for anyone delving into the world of mathematical analysis and its applications.

The Problem at Hand

Alright, now that we've got the basics covered, let's tackle the problem we're facing today. We're given two functions:

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

Our mission, should we choose to accept it (and I hope you do!), is to find $(f-g)(x)$ .

Solving for (f-g)(x)

Okay, here's where the fun begins! Remember that $(f-g)(x) = f(x) - g(x)$ . So, all we need to do is substitute the expressions for $f(x)$ and $g(x)$ into this equation:

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

Now, we need to simplify this expression. The key here is to distribute the negative sign in front of the second parenthesis:

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

Notice how the $- (4x - 2)$ became $-4x + 2$ . This is a crucial step! Mistakes in distributing the negative sign are common, so always double-check your work here. Now, let's combine like terms:

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

$(f-g)(x) = -7x - 3$

And that's it! We found that $(f-g)(x) = -7x - 3$ .

When solving this type of problem, it's super important to pay attention to the signs. Distributing the negative sign correctly is often the trickiest part. Also, remember to combine only the like terms. In this case, we combined the 'x' terms ( $-3x$ and $-4x$ ) and the constant terms ( $-5$ and $+2$ ) separately. This approach ensures that we arrive at the correct simplified expression for $(f-g)(x)$ . Always double-check your work, especially when dealing with negative signs and distribution, to avoid common errors and achieve accurate results. This methodical approach not only helps in solving the problem correctly but also builds a strong foundation for tackling more complex mathematical challenges involving function operations.

Why This Matters

You might be wondering, "Okay, we found $(f-g)(x)$ , but why should I care?" Well, understanding how to combine functions like this is super useful in many areas of math and science. For example, in physics, you might use this to find the net force acting on an object when you know the individual forces. In economics, you could use it to find the profit function by subtracting the cost function from the revenue function. These are just a couple of examples, but the possibilities are endless!

Let's Summarize

To recap, finding $(f-g)(x)$ involves these steps:

Understand that $(f-g)(x) = f(x) - g(x)$ .
Substitute the expressions for $f(x)$ and $g(x)$ into the equation.
Distribute the negative sign carefully.
Combine like terms.
Simplify the expression.

By following these steps, you'll be able to confidently tackle similar problems in the future. And remember, practice makes perfect! The more you work with functions, the easier it will become.

Common Mistakes to Avoid

Forgetting to Distribute the Negative Sign: This is probably the most common mistake. Always make sure to distribute the negative sign to every term inside the parentheses.
Combining Unlike Terms: You can only combine terms that have the same variable and exponent. For example, you can combine $-3x$ and $-4x$ , but you can't combine $-3x$ and $-5$ .
Rushing Through the Problem: Take your time and double-check each step. It's better to be accurate than fast.

Real-World Applications

Understanding the subtraction of functions, like finding $(f-g)(x)$ , is incredibly valuable in various real-world applications. Consider a scenario in business where $f(x)$ represents the revenue generated by selling $x$ units of a product, and $g(x)$ represents the cost of producing those $x$ units. The function $(f-g)(x)$ then gives the profit made from selling $x$ units. This is crucial for businesses to determine their profitability and make informed decisions about production levels. In environmental science, $f(x)$ could represent the amount of pollution produced by a factory, while $g(x)$ represents the amount of pollution that is naturally broken down by the environment. The function $(f-g)(x)$ then indicates the net pollution added to the environment, helping scientists and policymakers assess the impact of industrial activities and devise strategies for mitigation. Even in everyday life, understanding function subtraction can be useful. For example, if $f(x)$ represents the total calories you consume in a day and $g(x)$ represents the calories you burn through exercise, then $(f-g)(x)$ gives your net calorie intake, which is essential for managing your weight and health. These examples illustrate the broad applicability of function subtraction in analyzing and solving problems across diverse fields.

Practice Problems

Want to test your understanding? Try these practice problems:

If $f(x) = 5x + 2$ and $g(x) = 2x - 1$ , find $(f-g)(x)$ .
If $f(x) = -2x^2 + 3x$ and $g(x) = x^2 - 5x + 4$ , find $(f-g)(x)$ .
If $f(x) = x^3 - 1$ and $g(x) = x + 2$ , find $(f-g)(x)$ .

Work through these problems, and you'll be a pro at finding $(f-g)(x)$ in no time! Remember, the key is to take your time, distribute the negative sign carefully, and combine like terms. Good luck, and happy calculating!

Solution

The correct answer is A. $(f-g)(x)=-7 x-3$