Adding Functions: A Step-by-Step Guide

Hey math enthusiasts! Today, we're going to dive into the world of functions and learn how to add them together. We'll be working with two specific functions, f(x) and g(x), and figuring out what their sum looks like. This is a fundamental concept in algebra, so pay close attention, and let's get started!

Understanding the Functions

First, let's get acquainted with our functions. We have:

• f(x) = (x - 16) / (x² + 6x - 40), with the condition that x cannot equal -10 or 4. This is because these values would make the denominator zero, which isn't allowed.

• g(x) = 1 / (x + 10), also with the condition that x can't be -10.

So, essentially, f(x) is a fraction with a quadratic expression in the denominator, and g(x) is a simpler fraction. Our goal is to find an expression that represents f(x) + g(x).

Now, before we jump into the addition, let's quickly factor the denominator of f(x). Factoring a quadratic expression is a crucial skill. It often simplifies the problem, making it easier to solve. The quadratic in f(x) is x² + 6x - 40. We're looking for two numbers that multiply to -40 and add up to 6. Those numbers are 10 and -4. So, we can rewrite the denominator as (x + 10)(x - 4). That changes our f(x) into (x - 16) / ((x + 10)(x - 4)). Seeing the factored form of the denominator helps us see common factors, which we'll need for adding the fractions.

Remember, when adding fractions, it's essential to have a common denominator. This is the cornerstone of fraction arithmetic. Finding the common denominator can sometimes seem tricky, but it's an important skill. It ensures that we are adding like terms.

Let's get down to business and figure out this problem. We'll walk through this step by step.

Finding a Common Denominator

To add f(x) and g(x), we need a common denominator. Notice that the denominator of f(x), after factoring, is (x + 10)(x - 4), and the denominator of g(x) is (x + 10). The least common denominator (LCD) will be (x + 10)(x - 4) because it includes all the factors from both denominators.

To make the denominators the same, we'll need to multiply g(x) by a clever form of 1. Specifically, we'll multiply both the numerator and denominator of g(x) by (x - 4). This doesn't change the value of g(x), but it does allow us to write it with the same denominator as f(x).

So, g(x) becomes:

*g(x) = (1 / (x + 10)) * ((x - 4) / (x - 4)) g(x) = (x - 4) / ((x + 10)(x - 4))

Now both functions, f(x) and the modified g(x), have the same denominator, which is (x + 10)(x - 4). This is exactly what we need in order to add the two functions. The use of common denominators is vital. It's the core of how you manipulate fractions, and it helps you get to the right answer. Now, we're ready to add the numerators.

Adding the Functions

Now that we have a common denominator, we can add the numerators of f(x) and the modified g(x). Remember, f(x) = (x - 16) / ((x + 10)(x - 4)) and g(x) = (x - 4) / ((x + 10)(x - 4)). Adding them together gives us:

f(x) + g(x) = ((x - 16) / ((x + 10)(x - 4))) + ((x - 4) / ((x + 10)(x - 4)))

Since the denominators are the same, we can simply add the numerators:

f(x) + g(x) = (x - 16 + x - 4) / ((x + 10)(x - 4))

Combining like terms in the numerator, we get:

f(x) + g(x) = (2x - 20) / ((x + 10)(x - 4))

This is our answer! We've successfully added the two functions. We have a simple result. We are now one step closer to solving our problem.

Simplifying the Expression (Optional)

Let's see if we can simplify this further. The numerator is 2x - 20. Can we factor anything out of that? Yes, we can factor out a 2, giving us 2(x - 10). So our expression becomes:

f(x) + g(x) = (2(x - 10)) / ((x + 10)(x - 4))

Now, let's consider the answer choices. We have to determine which option is the correct answer. The options are shown at the beginning of the problem. We want to identify the expression that matches the answer that we have just calculated. Although, the numerator and denominator cannot be reduced any further, it is still the correct answer. The process of simplification can be very helpful to find the right solution.

Matching with the Options

Let's look back at the answer choices you provided:

A. (2x - 20) / (x² + 6x - 40) B. (x - 15) / (x² + 7x - 30) C. (2x - 20) / (x² - 6x - 40)

Looking at option A, it appears to be the most relevant answer. The denominator can be rewritten in its factored form (x+10)(x-4). So the expression is the same as the equation we generated earlier. Note that x² + 6x - 40 is the same as (x + 10)(x - 4).

Option B does not have the same form, so it cannot be the answer. Option C also has an incorrect term in the denominator.

Therefore, the correct answer is option A.

Conclusion

And there you have it! We've successfully added the two functions, found a common denominator, combined the numerators, and simplified the expression. The key takeaways here are understanding how to find a common denominator, add fractions, and simplify the resulting expression. Keep practicing these steps, and you'll become a pro at adding functions!

This process is fundamental in algebra and is essential for more complex mathematical concepts. Adding functions is like adding any other fractions, with a few extra steps for finding the common denominators. Also, simplifying your expressions is important. It helps make your answers clear and concise.

Adding functions may seem tricky at first, but with practice, it becomes second nature. Remember to pay close attention to detail, especially when factoring and simplifying. Always double-check your work! Now go forth and conquer those function problems, you got this!

I hope this helps! If you have any more questions, feel free to ask. Keep up the great work, and happy learning!