Domain and Its Impact on the Range

Oct 13, 2025 by windailys.com 94 views

To find the range of the function $f(x) = 4x + 9$ , we need to understand the relationship between the input values (domain) and the output values (range). The function is a linear equation, where the output value is determined by multiplying the input value by 4 and then adding 9.

Domain and Its Impact on the Range

The domain $D = \{-4, -2, 0, 2\}$ consists of four specific input values. To find the range, we need to substitute each of these values into the function and calculate the corresponding output values.

Calculating the Output Values

Let's calculate the output values for each input value in the domain:

For $x = -4$ , $f(-4) = 4(-4) + 9 = -16 + 9 = -7$
For $x = -2$ , $f(-2) = 4(-2) + 9 = -8 + 9 = 1$
For $x = 0$ , $f(0) = 4(0) + 9 = 0 + 9 = 9$
For $x = 2$ , $f(2) = 4(2) + 9 = 8 + 9 = 17$

Determining the Range

The output values we calculated are the corresponding values in the range. Therefore, the range of the function $f(x) = 4x + 9$ given the domain $D = \{-4, -2, 0, 2\}$ is $R = \{-7, 1, 9, 17\}$ .

Comparing with the Options

Now, let's compare our calculated range with the options provided:

A. $R = \{-7, 1, 9, 17\}$ : This option matches our calculated range.
B. $R = \{-17, -9, -1, 17\}$ : This option does not match our calculated range.
C. $R = \{-7, -1, 9, 17\}$ : This option does not match our calculated range.
D. $R = \{-7, 1, 9, 21\}$ : This option does not match our calculated range.

Conclusion

Based on our calculations, the correct answer is:

A. $R = \{-7, 1, 9, 17\}$

Q: What is the range of a function?

A: The range of a function is the set of all possible output values it can produce for the given input values.

Q: How do I find the range of a linear function?

A: To find the range of a linear function, you need to substitute the input values into the function and calculate the corresponding output values.

Q: What is the domain of a function?

A: The domain of a function is the set of all possible input values it can accept.

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

A: The domain of a function is usually specified in the problem statement or can be determined by analyzing the function itself.

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

A: The domain and range of a function are related in that the output values (range) are determined by the input values (domain).

Q: Can the range of a function be infinite?

A: Yes, the range of a function can be infinite if the function is not bounded.

Q: Can the range of a function be a single value?

A: Yes, the range of a function can be a single value if the function is a constant function.

Q: How do I determine if a function is a constant function?

A: A function is a constant function if it always produces the same output value for any input value.

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, while the range of a function is the set of all possible output values.

Q: Can the domain and range of a function be the same?

A: Yes, the domain and range of a function can be the same if the function is a one-to-one function.

Q: What is a one-to-one function?

A: A one-to-one function is a function that maps each input value to a unique output value.

Q: How do I determine if a function is one-to-one?

A: A function is one-to-one if it passes the horizontal line test, meaning that no horizontal line intersects the graph of the function in more than one place.

Q: What is the significance of the range of a function?

A: The range of a function is significant because it determines the possible output values of the function, which can be useful in various applications.

Q: Can the range of a function be used to determine the domain?

A: No, the range of a function cannot be used to determine the domain. The domain and range are two separate concepts that are related but distinct.

Q: How do I use the range of a function to solve problems?

A: The range of a function can be used to solve problems by determining the possible output values of the function and using that information to make decisions or predictions.

Q: What are some common applications of the range of a function?

A: The range of a function has many applications in various fields, including physics, engineering, economics, and computer science.

Q: Can the range of a function be used to model real-world phenomena?

A: Yes, the range of a function can be used to model real-world phenomena, such as population growth, financial transactions, and physical systems.

Q: How do I use the range of a function to model real-world phenomena?

A: To use the range of a function to model real-world phenomena, you need to identify the input values that correspond to the real-world situation and use the function to determine the possible output values.