Problem 32 - Calculus

Question

Are the functions even, odd, or neither? $$f(x)=2 x$$

Step-by-Step Solution

Answer

The function \( f(x) = 2x \) is an odd function.

Step 1: Understand Function Symmetry

To determine if a function is even, odd, or neither, we need to check its symmetry. An even function satisfies \( f(x) = f(-x) \) and is symmetric with respect to the y-axis. An odd function satisfies \( f(-x) = -f(x) \) and is symmetric with respect to the origin.

Step 2: Substitute and Simplify

Substitute \(-x\) into the function. Given \( f(x) = 2x \), substitute to get \( f(-x) = 2(-x) = -2x \).

Step 3: Check for Even Function

Check if \( f(x) = f(-x) \). We have \( f(x) = 2x \) and \( f(-x) = -2x \). These are not equal, so the function is not even.

Step 4: Check for Odd Function

Check if \( f(-x) = -f(x) \). We have \( f(-x) = -2x \) and \(-f(x) = -2x \). These are equal, so the function is odd.

Step 5: Conclusion

Since \( f(-x) = -f(x) \), the function \( f(x) = 2x \) is an odd function.

Key Concepts

Even Function

Even functions have a special type of symmetry. To explore whether a function is even, you need to verify the mathematical condition:

• \( f(x) = f(-x) \)

This means that if you flip the input across the y-axis (change \( x \) into \( -x \)), the function yields the same output.
An example of an even function could be \( f(x) = x^2 \). When you substitute \( -x \) for \( x \), you get \( (-x)^2 = x^2 \). The outputs before and after substitution remain the same, confirming even symmetry.
A key characteristic of even functions is that they are symmetric with respect to the y-axis. This means you can "fold" the graph along the y-axis, and the two sides would match perfectly.

Odd Function

Odd functions, on the other hand, exhibit symmetry around the origin. This type of symmetry can be checked using the condition:

• \( f(-x) = -f(x) \)

For odd functions, if you switch \( x \) with \( -x \), then the entire function outcome flips its sign.
Consider the function \( f(x) = x^3 \). When you substitute \( -x \), it becomes \( f(-x) = (-x)^3 = -x^3 \), which is indeed \( -f(x) \), thus showing odd symmetry.
Odd functions have the unique trait of being rotationally symmetric about the origin. Imagine rotating the graph 180 degrees around the origin, and it will look the same as before the rotation.

Function Substitution

Function substitution is a powerful tool used to determine symmetry. It involves replacing the variable \( x \) with \( -x \) to observe the changes in the function’s output.
For example, if you start with \( f(x) = 2x \) and perform substitution, you would calculate \( f(-x) = 2(-x) = -2x \). The purpose is to find whether the result matches \( f(x) \), or \(-f(x)\), or if it's different.
This step is crucial in exploring whether a function is even, odd, or neither. Understanding how substitution affects the function provides insights into its symmetry type.

Symmetry with Respect to Origin

Symmetry with respect to the origin is a defining feature of odd functions. In simpler terms, if you can spin the graph of a function 180 degrees around the origin (the point where the x-axis and y-axis cross), and it matches its original position, the function is symmetric about the origin.
This symmetry is mathematically captured by the condition:

• \( f(-x) = -f(x) \)

For instance, take \( f(x) = 2x \). When substituting \( -x \), the function becomes \( f(-x) = -2x \). Comparing \( f(-x) \) to \(-f(x)\), since both are equal, \( f(x) = 2x \) shows symmetry with respect to the origin.
Visualizing this symmetry helps in grasping how the function behaves when its graph undergoes a 180-degree rotation.