Check the individual functions

Oct 13, 2025 by windailys.com 28 views

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Introduction

When dealing with combined functions, it's essential to find the domain of the resulting function. In this article, we will explore how to find the domain of a combined function, specifically when the functions are added together. We will use the given functions $f(x)=9-7x$ and $g(x)=-5x+5$ to demonstrate the process.

Understanding the Domain of a Function

The domain of a function is the set of all possible input values (x-values) for which the function is defined. In other words, it's the set of all possible x-values that the function can accept without resulting in an undefined or imaginary value.

Finding the Domain of a Combined Function

To find the domain of a combined function, we need to consider the domains of the individual functions and then find the intersection of those domains. This is because the combined function is only defined where both individual functions are defined.

Step 1: Find the Domain of Each Individual Function

Let's start by finding the domain of each individual function.

Domain of $f(x)=9-7x$

The function $f(x)=9-7x$ is a linear function, and it's defined for all real numbers. Therefore, the domain of $f(x)$ is the set of all real numbers, which can be represented as $(-\infty, \infty)$ .

Domain of $g(x)=-5x+5$

The function $g(x)=-5x+5$ is also a linear function, and it's defined for all real numbers. Therefore, the domain of $g(x)$ is also the set of all real numbers, which can be represented as $(-\infty, \infty)$ .

Step 2: Find the Intersection of the Domains

Since both individual functions are defined for all real numbers, the intersection of their domains is also the set of all real numbers. Therefore, the domain of the combined function $(f+g)(x)$ is also the set of all real numbers, which can be represented as $(-\infty, \infty)$ .

Conclusion

In conclusion, the domain of the combined function $(f+g)(x)$ when $f(x)=9-7x$ and $g(x)=-5x+5$ is the set of all real numbers, which can be represented as $(-\infty, \infty)$ . This is because both individual functions are defined for all real numbers, and the intersection of their domains is also the set of all real numbers.

Example

Let's consider an example to illustrate this concept. Suppose we want to find the domain of the combined function $(f+g)(x)$ when $f(x)=\frac{1}{x}$ and $g(x)=\frac{1}{x-1}$ . In this case, the domain of $f(x)$ is the set of all real numbers except $x=0$ , and the domain of $g(x)$ is the set of all real numbers except $x=1$ . Therefore, the intersection of their domains is the set of all real numbers except $x=0$ and $x=1$ .

Tips and Tricks

Here are some tips and tricks to help you find the domain of a combined function:

Check the individual functions: Before finding the domain of the combined function, make sure to check the domains of the individual functions.
Find the intersection of the domains: The domain of the combined function is the intersection of the domains of the individual functions.
Use interval notation: When representing the domain of a function, use interval notation to make it easier to read and understand.

Final Thoughts

Finding the domain of a combined function is an essential step in understanding the behavior of the function. By following the steps outlined in this article, you can find the domain of a combined function and gain a deeper understanding of the function's behavior. Remember to always check the individual functions and find the intersection of their domains to ensure that you have the correct domain for the combined function.

References

[1] "Algebra and Trigonometry" by Michael Sullivan
[2] "Calculus" by James Stewart
[3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton

FAQs

Q: What is the domain of a combined function? A: The domain of a combined function is the intersection of the domains of the individual functions.
Q: How do I find the domain of a combined function? A: To find the domain of a combined function, check the individual functions and find the intersection of their domains.
Q: What is the difference between the domain of a function and the range of a function? A: The domain of a function is the set of all possible input values (x-values) for which the function is defined, while the range of a function is the set of all possible output values (y-values) that the function can produce.

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Introduction

In our previous article, we discussed how to find the domain of a combined function. However, we understand that there may be some questions and concerns that you may have. In this article, we will address some of the frequently asked questions (FAQs) related to the domain of a combined function.

Q&A

Q: What is the domain of a combined function?

A: The domain of a combined function is the set of all possible input values (x-values) for which the function is defined. In other words, it's the set of all possible x-values that the function can accept without resulting in an undefined or imaginary value.

Q: How do I find the domain of a combined function?

A: To find the domain of a combined function, you need to check the individual functions and find the intersection of their domains. This is because the combined function is only defined where both individual functions are defined.

Q: What is the difference between the domain of a function and the range of a function?

A: The domain of a function is the set of all possible input values (x-values) for which the function is defined, while the range of a function is the set of all possible output values (y-values) that the function can produce.

Q: Can the domain of a combined function be empty?

A: Yes, the domain of a combined function can be empty. This occurs when the intersection of the domains of the individual functions is empty.

Q: How do I represent the domain of a function in interval notation?

A: To represent the domain of a function in interval notation, you need to use the following notation:

(-∞, a): The domain is all real numbers less than a.
(a, ∞): The domain is all real numbers greater than a.
(-∞, a]: The domain is all real numbers less than or equal to a.
[a, ∞): The domain is all real numbers greater than or equal to a.
(-∞, ∞): The domain is all real numbers.

Q: Can the domain of a function be a single point?

A: Yes, the domain of a function can be a single point. This occurs when the function is defined only at that point.

Q: How do I find the domain of a function with a square root?

A: To find the domain of a function with a square root, you need to ensure that the expression inside the square root is non-negative. This means that the expression must be greater than or equal to zero.

Q: Can the domain of a function be a union of intervals?

A: Yes, the domain of a function can be a union of intervals. This occurs when the function is defined on multiple intervals.