Odd Function Graphs
Introduction
An odd function is a type of mathematical function that has a special symmetry property. Odd functions have the characteristic that when you replace the input variable ` x ` with ` -x `, the function returns its negative value. For example, if you put ` -x ` into the function and it equals ` -f(x) `, then the function is odd. Think of it like a mirror image across the origin. Some common examples of odd functions include the sine function, tangent function, and cosecant function. These functions exhibit symmetry about the origin. Understanding odd functions helps us analyze mathematical patterns and relationships, especially in areas like power functions and Fourier series. They're important because they help us understand how functions behave in different scenarios.
What Is an Odd Function?
A function `f(x)` is odd if `f(-x) = -f(x)` for all `x` and `-x` values in the domain of function `f`. This means that for every value of ` x ` if you put ` -x ` into the function, it gives you the negative of what you get when you put ` x ` in. The graph of an odd function is symmetrical about the origin, which means that it’s graph remains unchanged after a rotation of `180` degrees about the origin. For instance, ` f(x) = x^3 ` is an example of an odd function because it satisfies the condition ` -f(x) = f(-x) ` for all values of ` x `. This means that the function's behavior is the same whether you use ` x ` or ` -x `, just with opposite signs.
Understanding odd function definition helps in understanding symmetry and patterns in mathematical functions.
Odd Function Formula
The formula for identifying an odd function is straightforward:
`f(x)` is odd if `f(-x) = -f(x)`
In simpler terms, for every ` x ` value you put into the function, if you replace ` x ` with ` -x ` and the function equals the negative of what you started with, then it's an odd function.
Alternatively, an odd function satisfies the equation:
\( f(-x) + f(x) = 0 \)
This equation holds for all ` x ` values in the domain of the function. In essence, when you add the function's value at ` -x ` to its value at ` x `, you get zero for every ` x `. This property defines the symmetry of an odd function about the origin.
Graph of Odd Function
Odd functions exhibit a unique symmetry when graphed. They are symmetrical about the origin, meaning that if you were to reflect one side of the graph across the origin, you would get the other side. The graph of the function remains unchanged after a rotation of `180` degrees about the origin.
For example, the function ` y = x^3 ` is an odd function, and its graph reflects this symmetry about the origin.
We can state the same thing by saying that if the graph of an odd function is reflected over both axes, the result would be the original graph. If the graph of `f(x) = x^3` was first reflected over `y`-axis and then reflected over `x`-axis, the subsequent vertical and horizontal reflections would produce the original graph.
Other examples include functions like ` f(x) = -x ` and ` f(x) = 6\sin(x) `, where you can observe this symmetry. When you plot these odd functions, you'll notice that they maintain this symmetrical pattern about the origin.
Properties of Odd Functions
- Sum and Difference: Adding or subtracting two odd functions results in another odd function.
- Product and Quotient: Multiplying or dividing two odd functions produces an even function.
- Composition: Composing two odd functions yields an odd function.
- Uniqueness: Functions that are both even and odd are zero everywhere, and the absolute value of an odd function is even.
- Derivative and Integral: The derivative of an odd function is even, and the integral over a symmetric interval is always zero.
- Maclaurin Series: The Maclaurin series of any odd function contains only odd powers.
Odd Function versus Even Function
Odd function is the one in which the sign of the output gets changed if the sign of the input is changed while the output value remains the same. Whereas, an even function is the one which doesn’t have any change in output if the sign of input is changed.
In other words we can say that while an odd function demonstrates the relation `f(-x) = -f(x)`, an even function demonstrates the relation `f(-x) = f(x)`.
Solved Examples
Example `1`: Determine whether the function ` f(x) = 4x^3 + 5x ` is odd or not.
Solution:
To determine the nature of the function, we'll evaluate ` f(-x) ` and ` -f(x) `.
Given function: ` f(x) = 4x^3 + 5x `
\( f(-x) = 4(-x)^3 + 5(-x) \)
\( = -4x^3 - 5x \)
\( -f(x) = -(4x^3 + 5x) \)
\( = -4x^3 - 5x \)
Since ` f(-x) = -f(x) `, the function is odd.
Example `2`: Determine if the function ` f(x) = 2x^2 - 3x + 4 ` is odd or not.
Solution:
Let's find ` f(-x) ` and ` -f(x) ` to check the nature of the function.
Given function: ` f(x) = 2x^2 - 3x + 4 `
\( f(-x) = 2(-x)^2 - 3(-x) + 4 \)
\( = 2x^2 + 3x + 4 \)
\( -f(x) = -(2x^2 - 3x + 4) \)
\( = -2x^2 + 3x - 4 \)
Since ` f(-x) ` is not equal to `-f(x)`, the function is not an odd function.
Example `3`: State if the function represented by the below graph is odd or not.
Solution:
The graph is symmetric about the origin as for every point `(x, y)`, the corresponding point `(-x, -y)` is also on the graph. For example point `(1, 3)` is on the graph and the corresponding point `(-1, -3)` is also on the graph.
Example `4`: Prove that the function ` f(x) = x^5 + 4x^3 - x ` is an odd function.
Solution:
Let's calculate ` f(-x) ` and ` -f(x) ` to identify the nature of the function.
Given function: ` f(x) = x^5 + 4x^3 - x `
\( f(-x) = (-x)^5 + 4(-x)^3 - (-x) \)
\( = -x^5 - 4x^3 + x \)
\( -f(x) = -(x^5 + 4x^3 - x) \)
\( = -x^5 - 4x^3 + x \)
Since ` f(-x) = -f(x) `, the function is odd.
Example `5`: Check if the function ` f(x) = 2x^4 + 3x^2 - 1 ` odd or not.
Solution:
To determine the nature of the function, we'll evaluate ` f(-x) ` and ` -f(x) `.
Given function: ` f(x) = 2x^4 + 3x^2 - 1 `
\( f(-x) = 2(-x)^4 + 3(-x)^2 - 1 \)
\( = 2x^4 + 3x^2 - 1 \)
\( -f(x) = -(2x^4 + 3x^2 - 1) \)
\( = -2x^4 - 3x^2 + 1 \)
Since ` f(-x) ` is not equal to `-f(x)`, the function is not an odd function.
Practice Problems
Q`1`. If ` f(x) = x^2 `, which of the following statements is true?
- ` f(x) ` is neither even nor odd.
- ` f(x) ` is an even function.
- ` f(x) ` is an odd function.
- ` f(x) ` is both even and odd.
Answer: b
Q`2`. Consider the below graph. Does the graph represent an odd function or not?
- Yes
- No
- May be
- Cannot be determined
Answer: a
Q`3`. Determine the nature of the function ` f(x) = e^x `.
- Even
- Odd
- Neither even nor odd
- Cannot be determined
Answer: c
Q`4`. If ` f(x) = \sin(x) `, which of the following statements is correct?
- ` f(x) ` is neither even nor odd.
- ` f(x) ` is an even function.
- ` f(x) ` is an odd function.
- ` f(x) ` is both even and odd.
Answer: c
Q`5`. Is the function ` f(x) = x^4 - x^2 ` even, odd, or neither?
- Even
- Odd
- Neither even nor odd
- Cannot be determined
Answer: a
Frequently Asked Questions
Q`1`. What is an odd function?
Answer: An odd function is a mathematical function where ` f(-x) = -f(x) ` for all ` x ` in its domain. In simpler terms, if you replace ` x ` with its negative, the function returns the negative of its original value. Graphically, odd functions exhibit rotational symmetry about the origin.
Q`2`. What is an even function?
Answer: An even function is a mathematical function where ` f(x) = f(-x) ` for all ` x ` in its domain. In other words, if you replace ` x ` with its negative, the function's value remains unchanged. Graphs of even functions are symmetric about the `y`-axis.
Q`3`. How do I determine if a function is even, odd, or neither?
Answer: To determine the nature of a function, evaluate ` f(-x) ` and ` -f(x) `. If ` f(-x) = f(x) `, the function is even. If ` f(-x) = -f(x) `, the function is odd. If neither condition holds true, the function is neither even nor odd.
Q`4`. Can a function be both even and odd?
Answer: No, a function cannot be both even and odd simultaneously. If a function satisfies the condition for evenness, it cannot satisfy the condition for oddness, and vice versa. However, a function can be neither even nor odd.
Q`5`. What are some even and odd function examples?
Answer: Examples of even functions include ` f(x) = x^2 ` and ` g(x) = \cos(x) `, as their graphs are symmetric about the `y`-axis. Examples of odd functions include ` h(x) = x^3 ` and ` k(x) = \sin(x) `, as their graphs exhibit rotational symmetry about the origin.