Understanding the Domain and Range Exclusions

Oct 13, 2025 by windailys.com 113 views

When dealing with functions, it's essential to understand the concept of domain and range. The domain of a function is the set of all possible input values (x-values) for which the function is defined, while the range is the set of all possible output values (y-values) that the function can produce. In this article, we will explore the domain and range of functions, focusing on the specific case where the domain excludes the value $x \neq 3$ and the range excludes the value $y \neq 2$ .

Understanding the Domain and Range Exclusions

To determine the domain and range of a function, we need to consider the values that make the function undefined or produce a specific output value. In this case, we are looking for a function where $x \neq 3$ and $y \neq 2$ . This means that the function should not be defined at $x = 3$ and should not produce the output value $y = 2$ .

Analyzing the Given Functions

We are given four functions to analyze:

A. $f(x)=\frac{2(x+5)}{(x-3)}$ B. $f(x)=\frac{(x-5)}{(x+3)}$ C. $f(x)=\frac{(x+5)}{(x-3)}$ D. $f(x)=\frac{2(x+5)}{(x+3)}$

We need to examine each function to determine if it meets the criteria of having a domain where $x \neq 3$ and a range where $y \neq 2$ .

Function A: $f(x)=\frac{2(x+5)}{(x-3)}$

Let's start by analyzing function A. This function has a denominator of $(x-3)$ , which means that the function is undefined when $x = 3$ . This satisfies the first condition of the domain exclusion. However, we need to examine the range to see if it excludes the value $y = 2$ .

To determine the range, we can consider the behavior of the function as $x$ approaches positive and negative infinity. As $x$ approaches positive infinity, the function approaches $2$ . As $x$ approaches negative infinity, the function also approaches $2$ . This suggests that the function may produce the output value $y = 2$ .

However, we can also consider the behavior of the function as $x$ approaches $3$ from the left and right. As $x$ approaches $3$ from the left, the function approaches negative infinity. As $x$ approaches $3$ from the right, the function approaches positive infinity. This suggests that the function does not produce the output value $y = 2$ .

Function B: $f(x)=\frac{(x-5)}{(x+3)}$

Now, let's analyze function B. This function has a denominator of $(x+3)$ , which means that the function is undefined when $x = -3$ . This does not satisfy the first condition of the domain exclusion, which requires $x \neq 3$ .

Function C: $f(x)=\frac{(x+5)}{(x-3)}$

Next, let's analyze function C. This function has a denominator of $(x-3)$ , which means that the function is undefined when $x = 3$ . This satisfies the first condition of the domain exclusion. However, we need to examine the range to see if it excludes the value $y = 2$ .

To determine the range, we can consider the behavior of the function as $x$ approaches positive and negative infinity. As $x$ approaches positive infinity, the function approaches positive infinity. As $x$ approaches negative infinity, the function also approaches negative infinity. This suggests that the function does not produce the output value $y = 2$ .

Function D: $f(x)=\frac{2(x+5)}{(x+3)}$

Finally, let's analyze function D. This function has a denominator of $(x+3)$ , which means that the function is undefined when $x = -3$ . This does not satisfy the first condition of the domain exclusion, which requires $x \neq 3$ .

Conclusion

After analyzing the given functions, we can conclude that only function A meets the criteria of having a domain where $x \neq 3$ and a range where $y \neq 2$ . This is because function A has a denominator of $(x-3)$ , which makes the function undefined when $x = 3$ . Additionally, the behavior of the function as $x$ approaches positive and negative infinity suggests that it does not produce the output value $y = 2$ .

Final Answer

$The final answer is function A: $f(x)=\frac{2(x+5)}{(x-3)}$.<br/> **Domain and Range of Functions: Frequently Asked Questions**$

In our previous article, we explored the domain and range of functions, focusing on the specific case where the domain excludes the value $x \neq 3$ and the range excludes the value $y \neq 2$ . We analyzed four functions and determined that only function A meets the criteria. In this article, we will answer some frequently asked questions related to the domain and range of functions.

Q: What is the domain 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. In other words, it is the set of all possible values of x that can be plugged into the function.

Q: What is the range of a function?

A: The range of a function is the set of all possible output values (y-values) that the function can produce. In other words, it is the set of all possible values of y that can be obtained by plugging in different values of x.

Q: How do I determine the domain of a function?

A: To determine the domain of a function, you need to consider the values that make the function undefined. This can include values that make the denominator of the function equal to zero, as well as any other values that may cause the function to be undefined.

Q: How do I determine the range of a function?

A: To determine the range of a function, you need to consider the behavior of the function as x approaches positive and negative infinity. You can also consider the behavior of the function as x approaches specific values, such as x = 3 in the case of function A.

Q: What is the difference between the domain and 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 is the set of all possible output values (y-values) that the function can produce. In other words, the domain tells you what values of x you can plug into the function, while the range tells you what values of y you can expect to get out.

Q: Can a function have a domain that excludes a single value?

A: Yes, a function can have a domain that excludes a single value. For example, the function f(x) = 1 / (x - 3) has a domain that excludes the value x = 3, because the function is undefined when x = 3.

Q: Can a function have a range that excludes a single value?

A: Yes, a function can have a range that excludes a single value. For example, the function f(x) = 2x has a range that excludes the value y = 0, because the function never produces the output value y = 0.

Q: How do I graph a function with a domain that excludes a single value?

A: To graph a function with a domain that excludes a single value, you can use a hole in the graph to represent the excluded value. For example, if the function f(x) = 1 / (x - 3) has a domain that excludes the value x = 3, you can graph the function with a hole at x = 3.

Q: How do I graph a function with a range that excludes a single value?

A: To graph a function with a range that excludes a single value, you can use a vertical line to represent the excluded value. For example, if the function f(x) = 2x has a range that excludes the value y = 0, you can graph the function with a vertical line at y = 0.

Conclusion

In this article, we answered some frequently asked questions related to the domain and range of functions. We discussed how to determine the domain and range of a function, as well as how to graph a function with a domain or range that excludes a single value. We hope this article has been helpful in understanding the domain and range of functions.