Question

True or False A function \(f\) is decreasing on an interval 1 if, for any choice of \(x_{1}\) and \(x_{2}\) in \(1,\) with \(x_{1}f\left(x_{2}\right)\)

Short answer

True

Step-by-step solution

Understanding the Definition of Decreasing Function

A function is said to be decreasing on an interval if, for any two values \(x_1\) and \(x_2\) in that interval, whenever \(x_1 < x_2\), the function value at \(x_1\) is greater than the function value at \(x_2\). In other words, \(f(x_1) > f(x_2)\).

Analyzing the Given Statement

The given statement says that a function \(f\) is decreasing on an interval if for any choice of \(x_1\) and \(x_2\) in the interval 1, with \(x_1 < x_2\), then \(f(x_1) > f(x_2)\).

Checking the Consistency

Compare the given statement with the formal definition. Both the formal definition and the given statement assert that a function is decreasing if for any \(x_1 < x_2\) in the interval, \(f(x_1) > f(x_2)\).

Conclusion

Since the given statement matches the formal definition of a decreasing function, the statement is true.

Key concepts

Function

Visualizing functions can be done using graphs. The input values \(x\) are plotted on the horizontal axis (x-axis), and the output values \(f(x)\) are plotted on the vertical axis (y-axis). This visual representation can help to understand the behavior of the function, such as whether it is increasing, decreasing, or constant.

Interval

• Open interval: (a, b)
• Closed interval: [a, b]
• Half-open intervals: (a, b] or [a, b)

Knowing the type of interval is important for understanding which values are included and how they affect the behavior of functions over that range.

Definition

A function \(f\) is said to be decreasing on an interval if for any two values \(x_1\) and \(x_2\) where \(x_1 < x_2\), the function value at \(x_1\) is greater than the function value at \(x_2\), i.e., \(f(x_1) > f(x_2)\). This definition helps in identifying and analyzing functions based on their behavior within specified intervals.

Inequality

In the given exercise, the inequality \(f(x_1) > f(x_2)\) indicates that as we choose any two points \(x_1\) and \(x_2\) such that \(x_1 < x_2\), the function value at \(x_1\) will always be greater than the function value at 𝑓(𝑥\textsubscript{2}). By leveraging inequalities, we can characterize and make conclusions about the behavior of mathematical functions over specific intervals.