Problem 18 - The Calculus with Analytic Geometry

Question

There is one function that is both even and odd. What is it?

Step-by-Step Solution

Answer

The function that is both even and odd is \( f(x) = 0 \).

Step 1: Definition of Even Function

An even function satisfies the condition: \[ f(x) = f(-x) \]This means the function's value is the same for any positive or negative value of x.

Step 2: Definition of Odd Function

An odd function satisfies the condition: \[ f(x) = -f(-x) \]This means the function's value is the opposite for any positive or negative value of x.

Step 3: Determine Conditions for Both

To be both even and odd, a function must satisfy both conditions: \[ f(x) = f(-x) \] and \[ f(x) = -f(-x) \]Equating the two conditions gives: \[ f(x) = -f(x) \]

Step 4: Solve the Equation

From the equation \[ f(x) = -f(x) \]we can simplify to: \[ 2f(x) = 0 \]which results in: \[ f(x) = 0 \]

Step 5: Conclusion

The only function that satisfies both conditions, being both even and odd, is the function \( f(x) = 0 \). This is because it meets both criteria simultaneously.

Key Concepts

Even Function Definition

An even function is a mathematical function that exhibits symmetry about the y-axis. This means that the function has the same value at positive and negative inputs.
The formal definition: \( f(x) = f(-x) \).
This equation states that for every x, the function's value is equal to the value at -x.
Examples of even functions: \( f(x) = x^2 \), \( f(x) = \cos(x) \).
In a graphical representation, an even function will look the same on the left and right sides of the y-axis.

Odd Function Definition

An odd function is a mathematical function that exhibits symmetry about the origin. This means the function's value at any positive x is the negative of its value at negative x.
The formal definition: \( f(x) = -f(-x) \).
This equation means that for every x, the function's value at x is the opposite of its value at -x.
Examples of odd functions: \( f(x) = x^3 \), \( f(x) = \sin(x) \).
Graphically, an odd function will have rotational symmetry around the origin.

Zero Function

The zero function, also known as the null function, is the simplest type of function in mathematics. It is defined as: \( f(x) = 0 \), for all values of x.
This means no matter the input, the output is always zero.
The zero function is unique because it is the only function that is both even and odd simultaneously.
This is due to the fact that it satisfies both definitions: \( f(x) = f(-x) \) and \( f(x) = -f(-x) \). In this case, zero remains zero whether you multiply it by positive or negative values.
Graphically, the zero function is a straight line along the x-axis.