Understanding Continuous Linear Functions

Oct 13, 2025 by magicalceylontours.com 25 views

Understanding Continuous Linear Functions

A continuous linear function is a type of function that can be represented by a linear equation in the form of f(x) = mx + b, where m is the slope and b is the y-intercept. These functions are characterized by their ability to be graphed as a straight line, and they are continuous, meaning that there are no gaps or breaks in the graph.

Analyzing the Given Function

The given function, f(x), is a continuous linear function. To determine the interval over which the function is positive, we need to examine the values of the function at different points in its domain.

x	f(x)
-3
0
3
6
9
12

Determining the Slope and Y-Intercept

To determine the slope and y-intercept of the function, we can use the values of the function at two points. Let's use the points (0, f(0)) and (3, f(3)).

f(0) = 0 f(3) = 27

Using the formula for the slope, m = (f(3) - f(0)) / (3 - 0), we can calculate the slope:

m = (27 - 0) / (3 - 0) m = 27 / 3 m = 9

Now that we have the slope, we can use the point-slope form of a linear equation to find the y-intercept, b. The point-slope form is given by:

f(x) = m(x - x1) + f(x1)

Using the point (0, f(0)) = (0, 0), we can rewrite the equation as:

f(x) = 9(x - 0) + 0 f(x) = 9x

Finding the Interval Where the Function is Positive

Now that we have the equation for the function, f(x) = 9x, we can determine the interval over which the function is positive. To do this, we need to find the x-values for which f(x) > 0.

Since the function is a linear function, it will be positive for all x-values greater than the x-intercept. To find the x-intercept, we can set f(x) = 0 and solve for x:

9x = 0 x = 0

Since the x-intercept is at x = 0, the function will be positive for all x-values greater than 0.

Conclusion

In conclusion, the interval over which the function f(x) = 9x is positive is (0, ∞). This means that the function will be positive for all x-values greater than 0.

Key Takeaways

A continuous linear function is a type of function that can be represented by a linear equation in the form of f(x) = mx + b.
The slope and y-intercept of a linear function can be determined using the values of the function at two points.
The interval over which a linear function is positive can be determined by finding the x-intercept and considering all x-values greater than the x-intercept.

Final Answer

$The final answer is $\boxed{(0, ∞)}$.<br/> **Q&A: Selecting the Correct Interval for a Continuous Linear Function**$

Frequently Asked Questions

In this article, we will answer some of the most frequently asked questions related to selecting the correct interval for a continuous linear function.

Q: What is a continuous linear function?

A: A continuous linear function is a type of function that can be represented by a linear equation in the form of f(x) = mx + b, where m is the slope and b is the y-intercept. These functions are characterized by their ability to be graphed as a straight line, and they are continuous, meaning that there are no gaps or breaks in the graph.

Q: How do I determine the slope and y-intercept of a linear function?

A: To determine the slope and y-intercept of a linear function, you can use the values of the function at two points. Let's use the points (0, f(0)) and (3, f(3)). Using the formula for the slope, m = (f(3) - f(0)) / (3 - 0), you can calculate the slope. Then, using the point-slope form of a linear equation, you can find the y-intercept, b.

Q: How do I find the interval over which a linear function is positive?

A: To find the interval over which a linear function is positive, you need to find the x-values for which f(x) > 0. Since the function is a linear function, it will be positive for all x-values greater than the x-intercept. To find the x-intercept, you can set f(x) = 0 and solve for x.

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

A: The x-intercept of a linear function is the x-value at which the function crosses the x-axis. This is the point where the function is equal to zero.

Q: How do I determine the interval over which a linear function is negative?

A: To determine the interval over which a linear function is negative, you need to find the x-values for which f(x) < 0. Since the function is a linear function, it will be negative for all x-values less than the x-intercept.

Q: Can a linear function be both positive and negative?

A: Yes, a linear function can be both positive and negative. This occurs when the function has a positive slope and crosses the x-axis at a point other than the origin.

Q: How do I graph a linear function?

A: To graph a linear function, you can use the slope and y-intercept to plot two points on the graph. Then, draw a straight line through the two points to represent the function.

Q: What are some common applications of linear functions?

A: Linear functions have many real-world applications, including modeling population growth, calculating the cost of goods, and determining the distance between two points.

Q: Can linear functions be used to model non-linear relationships?

A: While linear functions can be used to model some non-linear relationships, they are not always the best choice. In many cases, more complex functions such as quadratic or exponential functions are needed to accurately model the relationship.

Conclusion

In conclusion, selecting the correct interval for a continuous linear function requires a thorough understanding of the function's properties, including its slope and y-intercept. By following the steps outlined in this article, you can determine the interval over which a linear function is positive or negative and gain a deeper understanding of these important mathematical concepts.

Key Takeaways

A continuous linear function is a type of function that can be represented by a linear equation in the form of f(x) = mx + b.
The slope and y-intercept of a linear function can be determined using the values of the function at two points.
The interval over which a linear function is positive or negative can be determined by finding the x-intercept and considering all x-values greater than or less than the x-intercept.
Linear functions have many real-world applications, including modeling population growth, calculating the cost of goods, and determining the distance between two points.

Final Answer

The final answer is $\boxed{(0, ∞)}$ .