Type The Correct Answer In Each Box. Use Numerals Instead Of Words.The Domain Of This Function Is $\{-12, -6, 3, 15\}$.Given The Function $y = -\frac{2}{3} X + 7$, Complete The Table Based On The Given

Understanding the Domain of a Function

The domain of a function is the set of all possible input 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 to get a valid output value. In this case, the domain of the function is given as $\{-12, -6, 3, 15\}$.

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

The given function is a linear function in the form $y = mx + b$, where m is the slope and b is the y-intercept. The slope of the function is $-\frac{2}{3}$, which means that for every 3 units we move to the right, the function value decreases by 2 units. The y-intercept is 7, which means that the function crosses the y-axis at the point (0, 7).

Completing the Table

To complete the table, we need to plug in each value of x from the domain into the function and calculate the corresponding value of y.

x	y
-12
-6
3
15

Calculating the Values of y

To calculate the value of y for each x, we can plug in the value of x into the function and simplify.

• For x = -12: $y = -\frac{2}{3}(-12) + 7$ $y = 8 + 7$ $y = 15$

• For x = -6: $y = -\frac{2}{3}(-6) + 7$ $y = 4 + 7$ $y = 11$

• For x = 3: $y = -\frac{2}{3}(3) + 7$ $y = -2 + 7$ $y = 5$

• For x = 15: $y = -\frac{2}{3}(15) + 7$ $y = -10 + 7$ $y = -3$

The Completed Table

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

Conclusion

In this article, we have completed the table for the given function $y = -\frac{2}{3} x + 7$ based on the domain $\{-12, -6, 3, 15\}$. We have calculated the value of y for each x by plugging in the value of x into the function and simplifying. The completed table shows the corresponding values of y for each x in the domain.

Key Takeaways

• The domain of a function is the set of all possible input values for which the function is defined.
• The function $y = -\frac{2}{3} x + 7$ is a linear function with a slope of $-\frac{2}{3}$ and a y-intercept of 7.
• To complete the table, we need to plug in each value of x from the domain into the function and calculate the corresponding value of y.
• The completed table shows the corresponding values of y for each x in the domain.

Further Reading

• Linear Functions: A linear function is a function that can be written in the form $y = mx + b$, where m is the slope and b is the y-intercept.
• Domain and Range: The domain of a function is the set of all possible input values for which the function is defined, while the range is the set of all possible output values.
• Calculus: Calculus is a branch of mathematics that deals with the study of rates of change and accumulation. It is used to solve problems in physics, engineering, and economics.
  Q&A: Domain and Range of a Linear Function =============================================

Frequently Asked Questions

Q: What is the domain of a function?

A: The domain of a function is the set of all possible input values for which the function is defined.

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

A: To find the domain of a function, you need to identify the values of x that make the function undefined. For example, if the function has a denominator of x, then x cannot be equal to 0.

Q: What is the range of a function?

A: The range of a function is the set of all possible output values for which the function is defined.

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

A: To find the range of a function, you need to identify the minimum and maximum values of the function. For example, if the function is a linear function, then the range is all real numbers.

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 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. For example, if the function is a linear function with a slope of 1 and a y-intercept of 0, then the domain and range are both all real numbers.

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

A: To determine if a function is a linear function, you need to check if it can be written in the form y = mx + b, where m is the slope and b is the y-intercept.

Q: What is the significance of the slope of a linear function?

A: The slope of a linear function represents the rate of change of the function. A positive slope means that the function is increasing, while a negative slope means that the function is decreasing.

Q: Can the slope of a linear function be zero?

A: Yes, the slope of a linear function can be zero. This means that the function is a horizontal line.

Q: How do I find the y-intercept of a linear function?

A: To find the y-intercept of a linear function, you need to plug in x = 0 into the function and solve for y.

Q: What is the significance of the y-intercept of a linear function?

A: The y-intercept of a linear function represents the point where the function crosses the y-axis.

Q: Can the y-intercept of a linear function be any value?

A: Yes, the y-intercept of a linear function can be any value. For example, if the function is y = 2x + 5, then the y-intercept is 5.

Q: How do I determine if a function is a linear function with a given domain?

A: To determine if a function is a linear function with a given domain, you need to check if the function can be written in the form y = mx + b, where m is the slope and b is the y-intercept, and the domain is the set of all possible input values.

Q: What is the significance of the domain of a linear function?

A: The domain of a linear function represents the set of all possible input values for which the function is defined.

Q: Can the domain of a linear function be any set of values?

A: Yes, the domain of a linear function can be any set of values. For example, if the function is y = 2x + 5, then the domain is all real numbers.

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

A: To find the range of a linear function with a given domain, you need to identify the minimum and maximum values of the function.

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

A: The range of a linear function represents the set of all possible output values for which the function is defined.

Q: Can the range of a linear function be any set of values?

A: Yes, the range of a linear function can be any set of values. For example, if the function is y = 2x + 5, then the range is all real numbers.

Q: How do I determine if a function is a linear function with a given range?

A: To determine if a function is a linear function with a given range, you need to check if the function can be written in the form y = mx + b, where m is the slope and b is the y-intercept, and the range is the set of all possible output values.

Q: What is the significance of the slope and y-intercept of a linear function?

A: The slope and y-intercept of a linear function represent the rate of change and the point where the function crosses the y-axis, respectively.

Q: Can the slope and y-intercept of a linear function be any values?

A: Yes, the slope and y-intercept of a linear function can be any values. For example, if the function is y = 2x + 5, then the slope is 2 and the y-intercept is 5.

Q: How do I find the domain and range of a linear function with a given slope and y-intercept?

A: To find the domain and range of a linear function with a given slope and y-intercept, you need to identify the set of all possible input values and the set of all possible output values.

Q: What is the significance of the domain and range of a linear function?

A: The domain and range of a linear function represent the set of all possible input values and the set of all possible output values, respectively.

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

A: Yes, the domain and range of a linear function can be the same. For example, if the function is y = 2x + 5, then the domain and range are both all real numbers.

Q: How do I determine if a function is a linear function with a given domain and range?

A: To determine if a function is a linear function with a given domain and range, you need to check if the function can be written in the form y = mx + b, where m is the slope and b is the y-intercept, and the domain and range are the set of all possible input values and the set of all possible output values, respectively.

Q: What is the significance of the slope and y-intercept of a linear function with a given domain and range?

A: The slope and y-intercept of a linear function with a given domain and range represent the rate of change and the point where the function crosses the y-axis, respectively.

Q: Can the slope and y-intercept of a linear function with a given domain and range be any values?

A: Yes, the slope and y-intercept of a linear function with a given domain and range can be any values. For example, if the function is y = 2x + 5, then the slope is 2 and the y-intercept is 5.

Q: How do I find the domain and range of a linear function with a given slope and y-intercept?

A: To find the domain and range of a linear function with a given slope and y-intercept, you need to identify the set of all possible input values and the set of all possible output values.

Q: What is the significance of the domain and range of a linear function with a given slope and y-intercept?

A: The domain and range of a linear function with a given slope and y-intercept represent the set of all possible input values and the set of all possible output values, respectively.

Q: Can the domain and range of a linear function with a given slope and y-intercept be the same?

A: Yes, the domain and range of a linear function with a given slope and y-intercept can be the same. For example, if the function is y = 2x + 5, then the domain and range are both all real numbers.

Q: How do I determine if a function is a linear function with a given domain and range?

A: To determine if a function is a linear function with a given domain and range, you need to check if the function can be written in the form y = mx + b, where m is the slope and b is the y-intercept, and the domain and range are the set of all possible input values and the set of all possible output values, respectively.

Q: What is the significance of the slope and y-intercept of a linear function with a given domain and range?

A: The slope and y-intercept of a linear function with a given domain and range represent the rate of change and the point where the function crosses the y-axis, respectively.

Q: Can the slope and y-intercept of a linear function with a given domain and range be any values?

A: Yes, the slope and y-intercept of a linear function with a given domain and range can be any values. For example, if the function is y = 2x + 5, then the slope is 2 and the y-intercept is 5.

Q: How do I find the domain and range of a linear function with a given slope and y-intercept?

A: To find the domain and range of a linear function with a given slope and y-intercept, you need to identify the set of all possible input values and the set of all possible output values.

Q: What is the significance of the domain and range of a linear function with a given slope and y-inter