Simplifying F(x) + G(x): A Step-by-Step Guide

Have you ever encountered complex functions and wondered how to simplify their addition? This article breaks down the process of simplifying the sum of two functions, f(x) and g(x). We'll walk through a specific example step-by-step, so you can confidently tackle similar problems. Specifically, we aim to simplify the expression f(x) + g(x), where f(x) = (x-16)/(x²+6x-40) and g(x) = 1/(x+10). This involves algebraic manipulation and understanding the domains of the functions involved. Let’s dive in!

Understanding the Functions

Before we jump into adding the functions, let’s understand them individually. Our first function is f(x) = (x-16)/(x²+6x-40). Notice that this is a rational function, a ratio of two polynomials. The denominator, x²+6x-40, is a quadratic expression. To fully understand f(x), we need to consider its domain. The domain of a rational function is all real numbers except for those values that make the denominator zero. These values are excluded because division by zero is undefined. Factoring the quadratic expression in the denominator will help us find these excluded values. The second function is g(x) = 1/(x+10), which is also a rational function. Its denominator is simpler, just x+10. Again, we need to be mindful of the domain of g(x). We'll explore how these domains affect the combined function later.

Deconstructing f(x) = (x-16)/(x²+6x-40)

Let's begin by thoroughly dissecting f(x). As highlighted earlier, the core of understanding this function lies in its denominator: x²+6x-40. This quadratic expression can be factored, which is a crucial step in simplifying the function and identifying its domain restrictions. Factoring the quadratic means finding two binomials that multiply together to give x²+6x-40. We look for two numbers that multiply to -40 and add to 6. These numbers are 10 and -4. Therefore, x²+6x-40 can be factored into (x+10)(x-4). Rewriting f(x) with the factored denominator gives us f(x) = (x-16)/((x+10)(x-4)). Now, we can clearly see the values of x that make the denominator zero: x = -10 and x = 4. These values must be excluded from the domain of f(x). The numerator, (x-16), doesn't directly affect the domain restrictions in this case but plays a vital role in the overall behavior of the function. It's important to keep track of both the numerator and the denominator when simplifying rational functions. This factorization also opens the door for potential simplifications if there are common factors between the numerator and the denominator, but in this instance, there are none. Understanding the factored form is also beneficial for sketching the graph of f(x) and analyzing its asymptotes and intercepts.

Examining g(x) = 1/(x+10)

Now, let’s turn our attention to g(x) = 1/(x+10). This function is a simpler rational function compared to f(x), but it's equally important to understand its behavior. The denominator of g(x) is simply (x+10). To determine the domain of g(x), we need to find the values of x that make the denominator zero. Setting x+10 equal to zero, we find that x = -10. This means that x = -10 is the only value that is excluded from the domain of g(x). The numerator of g(x) is simply 1, which means that the function will have a horizontal asymptote at y=0. The simplicity of g(x) makes it easier to analyze its graph and understand its behavior near the vertical asymptote at x = -10. It's important to recognize that the domain restriction of g(x) will also play a significant role when we add it to f(x). Specifically, since both functions have x ≠ -10 as a restriction, this restriction will apply to the sum of the functions as well. Understanding the individual characteristics of g(x) is crucial for correctly simplifying the combined expression f(x) + g(x). The interplay between the domains and the function behavior is a key aspect of working with rational functions.

Adding the Functions f(x) and g(x)

Now that we understand f(x) and g(x) individually, let’s add them together. We have f(x) = (x-16)/((x+10)(x-4)) and g(x) = 1/(x+10). To add fractions, they need to have a common denominator. In this case, the least common denominator (LCD) is (x+10)(x-4), which is already the denominator of f(x). We need to rewrite g(x) with this denominator. To do this, we multiply both the numerator and the denominator of g(x) by (x-4). This gives us g(x) = (1*(x-4))/((x+10)(x-4)) = (x-4)/((x+10)(x-4)). Now that both functions have the same denominator, we can add them. Adding the numerators, we get (x-16) + (x-4). The denominator remains (x+10)(x-4). Simplifying the numerator gives us 2x - 20. So, f(x) + g(x) = (2x - 20)/((x+10)(x-4)). This is a crucial step in simplifying the expression. We've combined the two functions into a single fraction. The next step will be to see if we can simplify this fraction further by factoring and canceling common factors.

Finding the Common Denominator

The crucial step in adding rational functions lies in identifying the common denominator. As we noted, we have f(x) = (x-16)/((x+10)(x-4)) and g(x) = 1/(x+10). The denominator of f(x) is (x+10)(x-4), and the denominator of g(x) is (x+10). To find the least common denominator (LCD), we need to consider all the factors present in both denominators. In this case, the factors are (x+10) and (x-4). The LCD must include each factor raised to the highest power that appears in any of the denominators. Here, (x+10) appears in both denominators, but only to the first power. The factor (x-4) appears only in the denominator of f(x), also to the first power. Therefore, the LCD is simply the product of these factors: (x+10)(x-4). This means that f(x) already has the common denominator, and we only need to adjust g(x). To get the common denominator for g(x), we multiply both its numerator and denominator by the missing factor, which is (x-4). This process ensures that we are creating an equivalent fraction, maintaining the value of the function while changing its form. Multiplying the numerator and denominator of g(x) by (x-4) is a fundamental algebraic technique for adding rational expressions. Once we have the common denominator, we can proceed with adding the numerators, which is the next step in simplifying the expression.

Combining the Numerators

With the common denominator established, we can now combine the numerators of f(x) and the modified g(x). Recall that we have f(x) = (x-16)/((x+10)(x-4)) and the modified g(x) = (x-4)/((x+10)(x-4)). Adding these functions involves adding their numerators while keeping the common denominator. So, f(x) + g(x) = ((x-16) + (x-4))/((x+10)(x-4)). The next step is to simplify the numerator by combining like terms. In the numerator, we have x and x, which combine to 2x. We also have -16 and -4, which combine to -20. Therefore, the simplified numerator is 2x - 20. The expression now becomes (2x - 20)/((x+10)(x-4)). This is a significant simplification, as we have combined the two original functions into a single rational expression. However, we're not done yet. The next step is to see if we can further simplify this expression by factoring and canceling common factors. Factoring the numerator, 2x - 20, can reveal potential cancellations with factors in the denominator. This step is crucial for obtaining the simplest form of the expression. The combination of numerators and simplification sets the stage for the final simplification steps.

Simplifying the Result

We’ve reached the point where f(x) + g(x) = (2x - 20)/((x+10)(x-4)). Now, let's see if we can simplify this expression further. The key to simplifying rational expressions is factoring. We can factor out a 2 from the numerator: 2x - 20 = 2(x - 10). The denominator is already in factored form: (x+10)(x-4). So, we have f(x) + g(x) = 2(x - 10)/((x+10)(x-4)). Now, we look for common factors in the numerator and the denominator. In this case, there are no common factors to cancel. The expression is now in its simplest form. However, it's important to remember the domain restrictions. Both f(x) and g(x) were undefined at x = -10, and f(x) was also undefined at x = 4. Therefore, the sum f(x) + g(x) is also undefined at these values. The final simplified expression is 2(x - 10)/((x+10)(x-4)), with the restrictions x ≠ -10 and x ≠ 4. This final step of checking for common factors and stating the domain restrictions is crucial for a complete and accurate simplification.

Factoring the Numerator

To simplify our expression (2x - 20)/((x+10)(x-4)), the first logical step is to factor the numerator. The numerator, 2x - 20, has a common factor of 2. We can factor out the 2, resulting in 2(x - 10). This factorization is a fundamental algebraic technique that allows us to identify potential cancellations with factors in the denominator. Factoring the numerator is crucial because it transforms the expression from a sum or difference into a product, which is essential for simplifying rational expressions. Once the numerator is factored, we can clearly see the factors present and compare them to the factors in the denominator. This comparison helps us determine if any cancellations are possible. In this specific case, the factored numerator, 2(x - 10), reveals a factor of (x - 10). We will then compare this to the factors in the denominator to see if any further simplification can be achieved. Factoring is a cornerstone of simplifying algebraic expressions, and it is particularly useful when dealing with rational functions.

Identifying and Canceling Common Factors

After factoring the numerator, we have the expression 2(x - 10)/((x+10)(x-4)). Now, we need to identify and cancel any common factors between the numerator and the denominator. In this case, the numerator has the factors 2 and (x - 10), while the denominator has the factors (x+10) and (x-4). Upon careful inspection, we can see that there are no common factors between the numerator and the denominator. This means that the expression is already in its simplest form and cannot be simplified further by canceling common factors. It's important to note that if we had found a common factor, such as (x - 4) in both the numerator and the denominator, we would cancel them out. This cancellation would result in a simplified expression, but it's crucial to remember that the original domain restrictions still apply. Since we don't have any common factors to cancel in this case, the expression 2(x - 10)/((x+10)(x-4)) remains as the simplified form. However, we still need to consider the domain restrictions to complete our simplification.

Stating the Domain Restrictions

Even after simplifying the expression to 2(x - 10)/((x+10)(x-4)), we're not quite finished. It's crucial to remember and state the domain restrictions. The original functions, f(x) and g(x), had denominators that could not be equal to zero. These restrictions carry over to the simplified expression. Looking back at the original functions, f(x) = (x-16)/((x+10)(x-4)) and g(x) = 1/(x+10), we identified that x cannot be -10 or 4. These values would make the denominators zero, leading to undefined expressions. Therefore, even though our simplified expression 2(x - 10)/((x+10)(x-4)) looks different, it still inherits these restrictions. We must state that x ≠ -10 and x ≠ 4. This is a critical step in working with rational functions because it ensures that we are only considering values of x for which the functions are defined. Stating the domain restrictions completes the simplification process and provides a comprehensive understanding of the function's behavior. The final answer is 2(x - 10)/((x+10)(x-4)), for x ≠ -10 and x ≠ 4. Remember, overlooking domain restrictions can lead to incorrect interpretations and applications of the function.

Conclusion

In summary, we successfully simplified the expression f(x) + g(x), where f(x) = (x-16)/(x²+6x-40) and g(x) = 1/(x+10). We found that f(x) + g(x) = 2(x - 10)/((x+10)(x-4)), with the crucial restrictions that x ≠ -10 and x ≠ 4. This process involved understanding the individual functions, finding a common denominator, combining numerators, factoring, and stating the domain restrictions. Each step is essential for a complete and accurate simplification. By understanding these steps, you can confidently tackle similar problems involving the addition and simplification of rational functions. Remember to always consider the domain restrictions, as they are a fundamental part of the function's definition.

For more information on rational functions and their simplification, you can visit Khan Academy's page on rational expressions. This resource provides additional examples and explanations to further enhance your understanding.