Great Lakes PGPM & PGDM 2025
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1 Questions around this concept.
Which of the following is an odd function?
Definition:
A function f(x) is said to be:
- Even if f(-x) = f(x) for all x in the domain.
- Odd if f(-x) = -f(x) for all x in the domain.
Characteristics:
- Graph of an even function is symmetric about the y-axis.
- Graph of an odd function is symmetric about the origin.
Example:
1. is an even function because
2. is an odd function because .
Foundation Building Questions:
Question 1: Determine whether the function is even, odd, or neither.
Solution:
Evaluate f(-x) :
Since f(-x) = f(x) , the function is even.
Question 2: Is the function even, odd, or neither?
Solution:
Evaluate g(-x) :
Since g(-x) = -g(x) , the function is odd.
Tips and Tricks:
1. Quick Check:
- For polynomial functions, if all the powers of x are even, then the function is likely even. If all the powers are odd, then the function is likely odd.
2. Graphical Insight:
- When in doubt, sketching a graph can quickly show if a function is symmetric about the y-axis (even) or the origin (odd).
3. Combination of Even and Odd:
- If a function has both even and odd powers of x , it's generally neither even nor odd. However, always verify using the definitions.
4. Value at Zero:
- Even functions always satisfy f(0) = f(-0) . For odd functions, f(0) is typically 0, but there are exceptions, so always cross-check.
Remember, understanding the symmetry and behaviour of functions can be a powerful tool, not just for solving direct questions but also for analysing the nature of functions in more complex scenarios. Practice by identifying the evenness or oddness of functions regularly.
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