ODD FUNCTION
Definition Of Odd Function
The function f(x) is said to be 'odd' if and only if f(x) is a real-valued function of a real variable x, and f(- x) = - f(x).
More About Odd Function
The graph of an odd function is symmetric with respect to the origin.
The product of two odd functions is even.
The product of an odd function and an even function is odd.
Example of Odd Function
The function f(x) = x3 is an odd function as:
f(- x) = (- x)3 = - x3 = - f(x)
So, f(- x) = - f(x).
Video Examples: Even and Odd Functions
Solved Example on Odd Function
Ques: If g(t) = sin h (9t), then which of the following is true?
Choices:
A. An odd function
B. An even function
C. Not an odd function
D. Neither an even nor an odd function
Correct Answer: A
Solution:
Step 1:g(t) = sin h (9t) = 
[Formula.]
Step 2:g(- 9t) = 
= 
Step 3: = - () = - g(t)
Step 4: Since g(- t) = - g(t), the given function is an odd function.
Quick Summary
- Graph is symmetric with respect to the origin.
- Product of two odd functions is even.
- Product of an odd function and an even function is odd.
🍎 Teacher Insights
Use graphical representations to illustrate the symmetry of odd functions. Emphasize the algebraic verification using the definition f(-x) = -f(x).🎓 Prerequisites
- Functions
- Real Numbers
- Symmetry
Check Your Knowledge
Q1: Which of the following functions is odd?
Q2: If g(t) = sinh(9t), then which of the following is true?
Frequently Asked Questions
Q: Is f(x) = 0 an odd function?
A: Yes, f(x) = 0 satisfies the condition f(-x) = -f(x) and is therefore considered an odd function.
Q: Are there functions that are neither even nor odd?
A: Yes, many functions do not exhibit either even or odd symmetry. For example, f(x) = x^2 + x.