Problem 83 - Calculus

Question

Are the statements true or false? Give an explanation for your answer. There is a function which is both even and odd.

Step-by-Step Solution

Answer

Yes, the zero function is both even and odd.

Step 1: Understand the Definitions

A function is called even if for all values of \( x \), \( f(-x) = f(x) \). A function is called odd if \( f(-x) = -f(x) \). To determine if a function can be both even and odd, we need to see if these two conditions can be satisfied simultaneously.

Step 2: Apply Definitions to Possible Functions

If a function is both even and odd, then it must satisfy both \( f(-x) = f(x) \) and \( f(-x) = -f(x) \). By equating these, we have \( f(x) = -f(x) \). This implies that \( 2f(x) = 0 \), leading to \( f(x) = 0 \).

Step 3: Evaluate the Zero Function

The function \( f(x) = 0 \) satisfies both the conditions for an even function \( f(-x) = f(x) \) and an odd function \( f(-x) = -f(x) \), since \( 0 = 0 \) and \( 0 = -0 \). Hence, the zero function is both even and odd.

Step 4: Determine the Generality

While the zero function is both even and odd, no other function apart from \( f(x) = 0 \) can satisfy both definitions since any other function would result in a contradiction from \( f(x) = -f(x) \).

Key Concepts

Function Definitions

Understanding the definitions of even and odd functions is crucial for solving related problems.
A function is considered **even** if its graph is symmetrical with respect to the y-axis. In mathematical terms, for any value of \(x\), the equation \(f(-x) = f(x)\) must hold true. This means that reflecting any point of the function across the y-axis will land on another point of the function.
On the other hand, a function is **odd** if its graph has rotational symmetry about the origin. This can be expressed mathematically as \(f(-x) = -f(x)\). This implies that if a point is rotated 180 degrees about the origin, it will coincide with another point of the function.
Knowing these definitions helps in identifying the nature of functions, which is fundamental in mathematical analysis.

Zero Function

The zero function, expressed as \(f(x) = 0\), is a special type of function where the output is constant and equals zero, regardless of the input.
- **Symmetry of the Zero Function**: The zero function is both even and odd. The conditions for a function to be even or odd boil down to symmetry rules:\(f(-x)=f(x)\) and \(f(-x)=-f(x)\).
- Specifically, for the zero function, \(f(x) = 0\), substituting in both conditions, simplifies to \(0 = 0\) for even and \(0 = -0\) for odd, both of which are true.
This means the zero function satisfies the properties of both evenness and oddness simultaneously, making it unique as no non-zero function can exhibit both these properties without resulting in contradictions.

Mathematical Proof

Let's delve into the mathematics that confirms a function cannot generally be both even and odd, except the zero function.
- **The Logic**: Since even functions abide by \(f(-x) = f(x)\) and odd functions by \(f(-x) = -f(x)\), they need to meet both conditions to be even and odd.
- **Simplification**: Equating both conditions yields \(f(x) = -f(x)\). By rearranging terms, we achieve \(2f(x) = 0\), giving us \(f(x) = 0\).
Thus, the zero function is the only function that satisfies both conditions, as it equates to zero for all values of \(x\). Through this small algebraic transformation, it becomes evident why the zero function is unique in this regard.

Function Properties

Different functions have distinct properties that can be understood through their symmetry and algebraic characteristics.
- **Even Functions**: These display symmetry across the y-axis and often appear prominently in quadratic functions or cosine trigonometric functions.
- **Odd Functions**: These exhibit rotational symmetry around the origin. Typically, cubic functions or sine functions fall into this category.
- **Uniqueness of the Zero Function**: It stands out as an exception, as it combines the properties of both even and odd functions due to its unvarying output of zero, irrespective of input. Recognizing these properties allows one to predict behavior and characteristics of functions in various mathematical contexts, giving insights into their deeper relationships and applications.