Question

If \(f(x)=\left\\{\begin{array}{cc}-x_{1} & -\pi

Short answer

The function is odd.

Step-by-step solution

Understand the Nature of the Function

The function \(f(x)\) is defined piecewise, with different expressions based on the range of \(x\). Specifically, for \(-\pi < x < 0\), \(f(x) = -x\), and for \(0 < x < \pi\), \(f(x) = x\).

Definition of Even and Odd Functions

A function \(f(x)\) is even if \(f(-x) = f(x)\) for all \(x\) in its domain. Similarly, a function is odd if \(f(-x) = -f(x)\) for all \(x\) in its domain.

Calculate f(-x) for Each Interval

Consider the interval \(-\pi < x < 0\). Here, \(f(x) = -x\). Let's find \(f(-x)\):- Since \(-x > 0\), use the second part of the piecewise function, so \(f(-x) = -x\).Now consider the interval \(0 < x < \pi\). Here, \(f(x) = x\). For \(f(-x)\):- Since \(-x < 0\), use the first part of the piecewise function, so \(f(-x) = x\).

Verify Odd Function Conditions

Since for \(-\pi < x < 0\), \(f(-x) = -x\) and \(-f(x) = -(-x) = x\) both match their respective parts of the function definition, and for \(0 < x < \pi\), \(f(-x) = x\) and \(-f(x) = -x\) also apply, it confirms the function adheres to the condition for being odd: \(f(-x) = -f(x)\).

Conclusion

The function \(f(x)\) satisfies the condition \(f(-x) = -f(x)\) across its entire domain in \((-\pi, \pi)\). Thus, \(f(x)\) is an odd function.

Key concepts

Piecewise Function

A piecewise function is a type of function that is defined by multiple sub-functions, with each sub-function applying to a specific interval of the main function's domain. This concept is crucial for creating flexibility in mathematical modeling, particularly when different rules or scenarios govern different parts of the domain.
In our exercise, the function \( f(x) \) is defined piecewise, with \( f(x) = -x \) for \( -\pi < x < 0 \) and \( f(x) = x \) for \( 0 < x < \pi \). This split allows the function to capture different behaviors on either side of the origin. Understanding when to apply each piece of the function is key to analyzing functions of this nature.
Piecewise functions are particularly useful because they allow complex real-world processes to be modeled mathematically. They can handle discontinuities, changes in trend, or other specialized constraints that a single, continuous function might not manage effectively.

Even and Odd Functions

Even and odd functions are categories that describe function symmetry with respect to the y-axis and origin, respectively. Recognizing whether a function is even or odd helps in understanding its symmetry properties, which are useful in graphing and solving equations.
To identify a function as even, it must hold the property \( f(-x) = f(x) \) for all \( x \) in its domain. This implies that the function's graph is mirrored over the y-axis.
In contrast, a function is odd if it satisfies \( f(-x) = -f(x) \) for all \( x \). This shows that the graph of the function exhibits rotational symmetry about the origin, meaning it looks the same even when rotated 180 degrees around the origin point. In this exercise, the piecewise function \( f(x) \) meets the criteria for an odd function.

Function Symmetry

Function symmetry offers insights into the behavior and visual representation of functions. Symmetrical properties can simplify the evaluation and integration of functions, as they often reveal patterns and consistency within the function's outputs.
Symmetry in functions can be recognized as either reflective symmetry across the y-axis (even functions) or rotational symmetry around the origin (odd functions). Rotational symmetry, as seen in odd functions, implies that every point \( (x, y) \) on the graph reflects as \( (-x, -y) \) if the function is odd.
Understanding these symmetry properties allows mathematicians to predict function behavior without needing to calculate every single point on its graph. Recognizing symmetry becomes a powerful tool not only for simplifying complex calculations but also in analyzing the underlying properties of equations simply and effectively.