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Oct 13, 2025 by windailys.com 56 views

Introduction

In mathematics, functions play a crucial role in modeling real-world phenomena. Understanding the behavior of functions is essential to analyze and solve problems in various fields, including physics, engineering, and economics. In this article, we will focus on the function q(x) = -|x-3|+4 and explore its behavior on different intervals.

The Function q(x) = -|x-3|+4

The function q(x) = -|x-3|+4 is a piecewise function, which means it has different expressions for different intervals of x. To understand the behavior of this function, we need to analyze its components.

Absolute Value Function

The absolute value function |x-3| is defined as:

|x-3| = { x-3, if x ≥ 3 -(x-3), if x < 3

This function has a minimum value of 0 at x = 3 and increases as x moves away from 3.

Linear Function

The linear function -|x-3|+4 is defined as:

-|x-3|+4 = { -|x-3|+4, if x ≥ 3 -|x-3|+4, if x < 3

This function has a maximum value of 4 at x = 3 and decreases as x moves away from 3.

Behavior of q(x) = -|x-3|+4

To understand the behavior of q(x) = -|x-3|+4, we need to analyze its components.

Increasing Interval

The function q(x) = -|x-3|+4 is increasing on the interval [3, ∞).

Decreasing Interval

The function q(x) = -|x-3|+4 is decreasing on the interval (-∞, 3).

Discussion

In this article, we have analyzed the behavior of the function q(x) = -|x-3|+4 on different intervals. We have shown that this function is increasing on the interval [3, ∞) and decreasing on the interval (-∞, 3).

Conclusion

In conclusion, understanding the behavior of functions is essential to analyze and solve problems in various fields. The function q(x) = -|x-3|+4 is a piecewise function that has different expressions for different intervals of x. By analyzing its components, we have shown that this function is increasing on the interval [3, ∞) and decreasing on the interval (-∞, 3).

Final Thoughts

In mathematics, functions play a crucial role in modeling real-world phenomena. Understanding the behavior of functions is essential to analyze and solve problems in various fields. By analyzing the behavior of functions, we can gain insights into the underlying mechanisms of the phenomena being modeled.

References

[1] "Functions" by Khan Academy
[2] "Piecewise Functions" by Math Open Reference
[3] "Absolute Value Functions" by Purplemath

Additional Resources

[1] "Functions" by Wolfram MathWorld
[2] "Piecewise Functions" by Mathway
[3] "Absolute Value Functions" by Symbolab

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Interval	Increasing	Decreasing
[3, ∞)
(-∞, 3)

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Introduction

In our previous article, we analyzed the behavior of the function q(x) = -|x-3|+4 on different intervals. In this article, we will answer some frequently asked questions about this function.

Q1: What is the domain of the function q(x) = -|x-3|+4?

A1: The domain of the function q(x) = -|x-3|+4 is all real numbers, i.e., (-∞, ∞).

Q2: What is the range of the function q(x) = -|x-3|+4?

A2: The range of the function q(x) = -|x-3|+4 is all real numbers less than or equal to 4, i.e., (-∞, 4].

Q3: On which interval is the function q(x) = -|x-3|+4 increasing?

A3: The function q(x) = -|x-3|+4 is increasing on the interval [3, ∞).

Q4: On which interval is the function q(x) = -|x-3|+4 decreasing?

A4: The function q(x) = -|x-3|+4 is decreasing on the interval (-∞, 3).

Q5: What is the maximum value of the function q(x) = -|x-3|+4?

A5: The maximum value of the function q(x) = -|x-3|+4 is 4, which occurs at x = 3.

Q6: What is the minimum value of the function q(x) = -|x-3|+4?

A6: The minimum value of the function q(x) = -|x-3|+4 is -∞, which occurs as x approaches -∞.

Q7: Is the function q(x) = -|x-3|+4 continuous?

A7: Yes, the function q(x) = -|x-3|+4 is continuous on its domain.

Q8: Is the function q(x) = -|x-3|+4 differentiable?

A8: Yes, the function q(x) = -|x-3|+4 is differentiable on its domain, except at x = 3.

Q9: What is the derivative of the function q(x) = -|x-3|+4?

A9: The derivative of the function q(x) = -|x-3|+4 is:

q'(x) = { -1, if x > 3 1, if x < 3

Q10: What is the second derivative of the function q(x) = -|x-3|+4?

A10: The second derivative of the function q(x) = -|x-3|+4 is:

q''(x) = { 0, if x > 3 0, if x < 3

Conclusion

In this article, we have answered some frequently asked questions about the function q(x) = -|x-3|+4. We hope that this article has provided you with a better understanding of this function and its behavior.

References

[1] "Functions" by Khan Academy
[2] "Piecewise Functions" by Math Open Reference
[3] "Absolute Value Functions" by Purplemath

Additional Resources

[1] "Functions" by Wolfram MathWorld
[2] "Piecewise Functions" by Mathway
[3] "Absolute Value Functions" by Symbolab