Domain Restrictions Of Composite Functions F(g(x)) Explained
In mathematics, understanding the domain of a function is crucial. The domain represents the set of all possible input values for which the function produces a valid output. When dealing with composite functions, such as f(g(x)), determining the domain requires careful consideration of the individual functions' restrictions and how they interact. This article delves into the process of finding the domain restrictions for the composite function f(g(x)), given f(x) = 1/(x+5) and g(x) = x-2. We will explore the step-by-step approach to identify any values of x that would make the composite function undefined.
To begin, let's define the functions we will be working with:
- f(x) = 1/(x+5)
- g(x) = x-2
The function f(x) is a rational function, which means it is a fraction where the numerator and denominator are polynomials. Rational functions have a significant restriction: the denominator cannot be zero. If the denominator were zero, the function would be undefined due to division by zero, which is mathematically impermissible. Therefore, for f(x), we need to ensure that x + 5 ≠0. Solving this inequality, we find that x ≠-5. This means that x = -5 is not in the domain of f(x).
The function g(x) = x-2 is a linear function. Linear functions are defined for all real numbers, meaning there are no restrictions on the input x. You can plug in any real number for x, and the function will produce a valid output. This characteristic of linear functions makes them straightforward to work with when considering composite functions.
The composite function f(g(x)) means we are plugging the function g(x) into the function f(x) wherever we see x. Let's perform this substitution:
f(g(x)) = f(x-2) = 1/((x-2)+5) = 1/(x+3)
So, the composite function f(g(x)) simplifies to 1/(x+3). This resulting function is again a rational function, similar to the original f(x). This means we need to apply the same restriction we discussed earlier: the denominator cannot be zero. Setting the denominator equal to zero and solving for x gives us the values that must be excluded from the domain.
Now that we have the composite function f(g(x)) = 1/(x+3), we need to find any restrictions on its domain. As we established, the denominator of a rational function cannot be zero. Thus, we need to find the values of x that make the denominator x+3 equal to zero. Setting x+3 = 0 and solving for x, we get:
x+3 = 0 x = -3
This means that x = -3 makes the denominator zero, and therefore, the function f(g(x)) is undefined at x = -3. This is a crucial restriction on the domain of the composite function. However, this is not the only restriction we need to consider.
When determining the domain of a composite function, it's not sufficient to only look at the final composite function. We also need to consider any restrictions imposed by the inner function, g(x), and the original function, f(x). In our case, g(x) = x - 2 has no restrictions, as it is a linear function defined for all real numbers. However, f(x) = 1/(x+5) has a restriction at x = -5. We must ensure that g(x) does not produce an output that would cause f(x) to be undefined.
To find this, we set g(x) equal to the restricted value of f(x), which is -5:
g(x) = -5 x - 2 = -5 x = -3
This result confirms our previous finding that x = -3 is a restriction. However, it highlights the importance of considering the original function's restrictions as well. In some cases, the inner function might introduce additional restrictions that are not immediately apparent from the composite function alone.
Considering all the restrictions, we have:
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The composite function f(g(x)) = 1/(x+3) has a restriction at x = -3. This is because x = -3 makes the denominator zero.
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The original function f(x) = 1/(x+5) has a restriction at x = -5. We need to ensure that g(x) does not produce an output of -5. In this specific case, solving g(x) = -5 also leads us to x = -3, which we already identified.
Therefore, the only restriction on the domain of f(g(x)) is x ≠-3. All other real numbers are permissible inputs for the composite function.
In conclusion, the domain of the composite function f(g(x)) = 1/(x+3), where f(x) = 1/(x+5) and g(x) = x-2, is restricted by the value x = -3. This restriction arises because this value makes the denominator of the composite function equal to zero, leading to an undefined result. When determining the domain of composite functions, it is essential to consider both the restrictions of the individual functions and the resulting composite function. This ensures a comprehensive understanding of the valid input values for the composite function.
The correct answer is B. x ≠-3.
This process highlights the importance of understanding the fundamental principles of functions and their domains in mathematics. By carefully analyzing the components of a composite function, we can accurately determine its restrictions and ensure the validity of mathematical operations.
Understanding the domain of a function is crucial in mathematics, as it defines the set of all possible input values for which the function produces a valid output. This becomes particularly important when dealing with composite functions, where one function is nested inside another, such as f(g(x)). Determining the domain restrictions for composite functions requires careful consideration of the individual functions' domains and how they interact. This article provides a comprehensive guide to identifying domain restrictions in composite functions, using the example f(g(x)) where f(x) = 1/(x+5) and g(x) = x-2. We will explore a step-by-step approach to identify any values of x that would make the composite function undefined, ensuring a thorough understanding of this fundamental concept.
Step 1: Analyzing the Individual Functions
To determine the domain of a composite function, we must first understand the domains of the individual functions involved. Let's start by analyzing the given functions:
- f(x) = 1/(x+5)
- g(x) = x-2
f(x) is a rational function, which means it is expressed as a fraction with polynomials in the numerator and denominator. The key characteristic of rational functions is that their denominator cannot be zero. Division by zero is undefined in mathematics, leading to a restriction on the domain. For f(x), the denominator is x + 5. To find the restriction, we set the denominator not equal to zero:
x + 5 ≠0 x ≠-5
This means that x = -5 is not in the domain of f(x). If x were to equal -5, the denominator would become zero, and f(x) would be undefined. Thus, the domain of f(x) is all real numbers except -5. Understanding this restriction is crucial for determining the overall domain of the composite function.
The function g(x) = x - 2 is a linear function. Linear functions have a straightforward domain: they are defined for all real numbers. There are no values of x that would make g(x) undefined. This is because linear functions do not involve division, square roots, or other operations that could lead to restrictions. The absence of restrictions in g(x) simplifies the process of finding the domain of the composite function, but we must still consider how it interacts with the restrictions of f(x).
Step 2: Constructing the Composite Function
Now that we understand the individual functions, we need to construct the composite function f(g(x)). This involves substituting g(x) into f(x) wherever we see x. The process is as follows:
f(g(x)) = f(x-2)
We replace the x in f(x) with the entire expression for g(x), which is x-2:
f(x-2) = 1/((x-2)+5)
Now, we simplify the expression:
1/((x-2)+5) = 1/(x-2+5) = 1/(x+3)
So, the composite function f(g(x)) simplifies to 1/(x+3). This resulting function is another rational function, similar to the original f(x). This means we need to apply the same restriction: the denominator cannot be zero. Recognizing that f(g(x)) is a rational function helps us anticipate and identify potential domain restrictions.
Step 3: Identifying Restrictions on the Composite Function
With the composite function f(g(x)) = 1/(x+3) in hand, we now focus on finding any restrictions on its domain. As with the original f(x), the primary restriction comes from the denominator. The denominator of f(g(x)) is x + 3. To find the values of x that must be excluded from the domain, we set the denominator not equal to zero:
x + 3 ≠0
Solving this inequality for x, we get:
x ≠-3
This tells us that x = -3 makes the denominator zero, and therefore, the function f(g(x)) is undefined at x = -3. This is a critical restriction on the domain of the composite function. However, it is essential to remember that this is not the only restriction we need to consider. We must also account for any restrictions imposed by the original functions, particularly the inner function, g(x), and how its output affects the outer function, f(x).
Step 4: Considering the Original Function Restrictions
When determining the domain of a composite function, it's not enough to only consider the final composite function. We must also consider the restrictions imposed by the individual functions that make up the composite. In our case, we have f(x) = 1/(x+5) and g(x) = x-2. We already established that f(x) has a restriction at x = -5, meaning x cannot be -5. The function g(x) = x-2 has no restrictions, as it is defined for all real numbers.
However, we need to ensure that g(x) does not produce an output that would cause f(x) to be undefined. In other words, we need to find any values of x that would make g(x) equal to -5, because that would cause the denominator of f(x) to be zero. To find this, we set g(x) equal to the restricted value of f(x):
g(x) = -5 x - 2 = -5
Solving this equation for x, we get:
x = -5 + 2 x = -3
This result is significant because it confirms our earlier finding that x = -3 is a restriction. It highlights the importance of considering the original function's restrictions as well. In some cases, the inner function might introduce additional restrictions that are not immediately apparent from the composite function alone. By considering both the composite function and the individual functions, we ensure a comprehensive understanding of the domain restrictions.
Step 5: Defining the Final Domain
Having analyzed all potential restrictions, we can now define the final domain of the composite function f(g(x)). We found that:
- The composite function f(g(x)) = 1/(x+3) has a restriction at x = -3. This is because x = -3 makes the denominator zero.
- The original function f(x) = 1/(x+5) has a restriction at x = -5. We needed to ensure that g(x) does not produce an output of -5. Solving g(x) = -5 led us to x = -3, which we already identified.
Therefore, the only restriction on the domain of f(g(x)) is x ≠-3. This means that all real numbers, except -3, are permissible inputs for the composite function. This thorough analysis ensures that we have accounted for all potential domain restrictions, leading to a clear and accurate understanding of the function's behavior.
Conclusion
In conclusion, determining the domain of a composite function requires a systematic approach that considers both the composite function itself and the individual functions that make it up. By analyzing the denominators, identifying potential restrictions, and ensuring that the inner function's output does not cause the outer function to be undefined, we can accurately determine the domain of the composite function. For the given functions f(x) = 1/(x+5) and g(x) = x-2, the domain of the composite function f(g(x)) = 1/(x+3) is all real numbers except x = -3. This comprehensive approach ensures that we avoid division by zero and other mathematical inconsistencies, leading to a robust understanding of the function's behavior.
The correct answer is B. x ≠-3.
Understanding composite function domains is essential in mathematics, and this guide provides the tools necessary to confidently tackle such problems. By following the steps outlined, students and practitioners can effectively navigate the complexities of composite functions and ensure accurate results.
Introduction: Understanding Function Domains
In the realm of mathematics, a function's domain is a fundamental concept, representing the set of all possible input values for which the function produces a valid output. Grasping the domain is crucial for analyzing function behavior and solving mathematical problems accurately. When dealing with composite functions, where one function is nested within another, determining the domain becomes a multi-faceted process. This article focuses on how to identify and understand domain restrictions in composite functions, using the specific example of f(g(x)) where f(x) = 1/(x+5) and g(x) = x-2. We will provide a clear, step-by-step methodology to determine any values of x that would render the composite function undefined, offering a comprehensive understanding of this essential mathematical concept.
Step 1: Analyzing the Domains of Individual Functions
The first step in understanding the domain of a composite function is to analyze the domains of the individual functions involved. This means identifying any values of x that would make each function undefined. For the given functions:
- f(x) = 1/(x+5)
- g(x) = x-2
Let's start with f(x). This function is a rational function, defined as a ratio of two polynomials. A key characteristic of rational functions is that their denominator cannot be equal to zero. Division by zero is undefined in mathematics and introduces a significant restriction on the function's domain. In f(x) = 1/(x+5), the denominator is (x+5). To find the restricted values, we set the denominator not equal to zero:
x + 5 ≠0
Solving for x, we get:
x ≠-5
Therefore, x = -5 is not in the domain of f(x). If x were to equal -5, the denominator would be zero, and f(x) would be undefined. Understanding this restriction is crucial for determining the domain of the composite function. Without this understanding, one might overlook a critical value that invalidates the function.
Now let's consider g(x) = x-2. This function is a linear function. Linear functions are characterized by their simple form and straightforward domains. Linear functions are defined for all real numbers, meaning there are no restrictions on the input x. You can input any real number for x, and the function will produce a valid real-number output. This is because linear functions do not involve operations such as division by a variable expression, square roots of variable expressions, or logarithms, which often lead to domain restrictions. The lack of restrictions in g(x) simplifies our analysis somewhat, but we still need to consider how its output interacts with the domain of f(x) in the composite function.
Step 2: Constructing the Composite Function f(g(x))
The next step is to construct the composite function f(g(x)). This involves substituting the entire function g(x) into the function f(x) wherever x appears. Let's perform this substitution step by step:
f(g(x)) = f(x-2)
Here, we replace the x in f(x) with the expression for g(x), which is (x-2). This gives us:
f(x-2) = 1/((x-2)+5)
Now, we simplify the expression inside the parentheses in the denominator:
1/((x-2)+5) = 1/(x-2+5) = 1/(x+3)
Thus, the composite function f(g(x)) simplifies to 1/(x+3). This resulting function is another rational function, much like the original f(x). This observation is significant because it alerts us to the potential for domain restrictions arising from the denominator. Specifically, we need to ensure that the denominator of the composite function, (x+3), is not equal to zero. Recognizing the structure of the composite function as rational guides our next steps in identifying domain restrictions.
Step 3: Identifying Initial Restrictions on the Composite Function
Now that we have the composite function f(g(x)) = 1/(x+3), our primary task is to identify any restrictions on its domain. As with any rational function, the key restriction arises from the denominator: it cannot be zero. Setting the denominator equal to zero will help us find the values of x that must be excluded from the domain. We set (x+3) not equal to zero:
x + 3 ≠0
Solving this inequality for x, we get:
x ≠-3
This result indicates that x = -3 makes the denominator of f(g(x)) equal to zero, making the function undefined at this point. This is a crucial restriction on the domain of the composite function. If we were to plug in x = -3 into f(g(x)), we would encounter division by zero, which is mathematically invalid. However, this restriction is not the only one we need to consider. Understanding this x ≠-3 restriction is an initial step, but we must delve deeper to ensure we have accounted for all possible restrictions imposed by the original functions.
Step 4: Considering the Domain of the Inner Function and its Impact
Determining the domain of a composite function requires more than just examining the final composite expression. We must also consider the domains of the individual functions that make up the composition. In our case, we have f(x) = 1/(x+5) and g(x) = x-2. We've already noted that f(x) has a restriction at x = -5, and g(x) has no restrictions on its own domain. However, the output of g(x) becomes the input of f(x) in the composite function f(g(x)). Therefore, we need to ensure that the output of g(x) does not equal -5, as that would lead to f(x) being undefined.
To determine any such restrictions, we set g(x) equal to the restricted value of f(x), which is -5:
g(x) = -5
Substituting the expression for g(x), we have:
x - 2 = -5
Solving this equation for x, we get:
x = -5 + 2 x = -3
This result is significant. It indicates that when x = -3, the output of g(x) is -5, which then becomes the input of f(x). Since f(x) is undefined at x = -5, this means that f(g(x)) is also undefined when x = -3. This confirms the restriction we found earlier by analyzing the composite function directly, but it reinforces the importance of considering the interaction between the inner and outer functions in a composition. This process ensures that we identify all possible restrictions on the domain, not just those immediately apparent from the composite function itself.
Step 5: Defining the Final Domain of the Composite Function
Having considered all potential sources of domain restrictions, we can now definitively state the domain of the composite function f(g(x)) = 1/(x+3). We found:
- The composite function f(g(x)) = 1/(x+3) has a restriction at x = -3, as it makes the denominator zero.
- The inner function g(x), when x = -3, produces an output of -5, which is the restricted value for f(x).
Therefore, the only restriction on the domain of f(g(x)) is x ≠-3. This means that the domain of the composite function includes all real numbers except -3. We have systematically addressed all potential restrictions, ensuring a complete and accurate determination of the domain.
Conclusion
In summary, determining the domain of a composite function requires a comprehensive approach. We must first identify any domain restrictions on the individual functions involved. Next, we construct the composite function and identify any additional restrictions that arise from its form. Finally, we must consider how the inner function's output interacts with the outer function's domain restrictions. For the functions f(x) = 1/(x+5) and g(x) = x-2, the composite function f(g(x)) = 1/(x+3) has a domain of all real numbers except x = -3. By following this step-by-step method, students and mathematicians can confidently determine the domain of composite functions, ensuring accuracy in mathematical analysis and problem-solving.
The correct answer is B. x ≠-3.
