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Oct 13, 2025 by magicalceylontours.com 103 views

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 is the set of all possible x-values that the function can accept without resulting in an undefined or imaginary output.

Given Function

The given function is:

${y = -\frac{2}{3}x + 7}$

This is a linear function in the form of y = mx + b, where m is the slope and b is the y-intercept.

Domain of the Function

The domain of the function is given as:

${\{-12, -6, 3, 15\}}$

This means that the function is defined for the x-values -12, -6, 3, and 15.

Completing the Table

To complete the table, we need to find the corresponding y-values for each x-value in the domain.

Step 1: Find the y-value for x = -12

Substitute x = -12 into the function:

${y = -\frac{2}{3}(-12) + 7}$

Simplify the expression:

${y = \frac{24}{3} + 7}$

${y = 8 + 7}$

${y = 15}$

Step 2: Find the y-value for x = -6

Substitute x = -6 into the function:

${y = -\frac{2}{3}(-6) + 7}$

Simplify the expression:

${y = \frac{12}{3} + 7}$

${y = 4 + 7}$

${y = 11}$

Step 3: Find the y-value for x = 3

Substitute x = 3 into the function:

${y = -\frac{2}{3}(3) + 7}$

Simplify the expression:

${y = -2 + 7}$

${y = 5}$

Step 4: Find the y-value for x = 15

Substitute x = 15 into the function:

${y = -\frac{2}{3}(15) + 7}$

Simplify the expression:

${y = -10 + 7}$

${y = -3}$

Completed Table

x	y
-12	15
-6	11
3	5
15	-3

Conclusion

The domain of the function is ${\{-12, -6, 3, 15\}}$ . We have completed the table by finding the corresponding y-values for each x-value in the domain. The completed table is shown above.

Key Takeaways

The domain of a function is the set of all possible input values (x-values) for which the function is defined.
The given function is a linear function in the form of y = mx + b.
We completed the table by finding the corresponding y-values for each x-value in the domain.

Practice Problems

Find the domain of the function y = 2x - 5.
Complete the table for the function y = x^2 - 4.
Find the y-value for x = 0 in the function y = 3x + 2.

Answer Key

The domain of the function is all real numbers.
The completed table is shown below:

x	y
-2	0
0	-4
2	0
4	4

The y-value for x = 0 is 2.
Type the Correct Answer in Each Box =====================================

Q&A: Domain of a Function

Q1: What is the domain of a function?

A1: The domain of a function is the set of all possible input values (x-values) for which the function is defined.

Q2: How do you find the domain of a function?

A2: To find the domain of a function, you need to identify any restrictions on the input values (x-values) that would result in an undefined or imaginary output.

Q3: What is the difference between the domain and range of a function?

A3: The domain of a function is the set of all possible input values (x-values), while the range of a function is the set of all possible output values (y-values).

Q4: Can a function have a domain of all real numbers?

A4: Yes, a function can have a domain of all real numbers. For example, the function y = 2x - 5 has a domain of all real numbers.

Q5: How do you determine the domain of a function with a square root?

A5: To determine the domain of a function with a square root, you need to ensure that the expression inside the square root is non-negative.

Q6: Can a function have a domain of a single value?

A6: Yes, a function can have a domain of a single value. For example, the function y = 2x - 5 has a domain of a single value, x = 0.

Q7: How do you find the domain of a function with a fraction?

A7: To find the domain of a function with a fraction, you need to ensure that the denominator is not equal to zero.

Q8: Can a function have a domain of a set of discrete values?

A8: Yes, a function can have a domain of a set of discrete values. For example, the function y = 2x - 5 has a domain of a set of discrete values, x = -2, 0, 2.

Q9: How do you determine the domain of a function with a logarithm?

A9: To determine the domain of a function with a logarithm, you need to ensure that the expression inside the logarithm is positive.

Q10: Can a function have a domain of a set of complex numbers?

A10: Yes, a function can have a domain of a set of complex numbers. For example, the function y = 2x - 5 has a domain of a set of complex numbers.

Conclusion

The domain of a function is an important concept in mathematics that determines the set of all possible input values (x-values) for which the function is defined. We have answered some common questions about the domain of a function, including how to find the domain, the difference between the domain and range, and how to determine the domain of a function with a square root, fraction, logarithm, or complex numbers.

Practice Problems

Find the domain of the function y = 2x - 5.
Complete the table for the function y = x^2 - 4.
Find the y-value for x = 0 in the function y = 3x + 2.

Answer Key

The domain of the function is all real numbers.
The completed table is shown below:

x	y
-2	0
0	-4
2	0
4	4

The y-value for x = 0 is 2.

Additional Resources

Khan Academy: Domain and Range
Mathway: Domain and Range
Wolfram Alpha: Domain and Range