Physics and Mathematics
Odd and Even Function
1. What is an Even Function?
A function f(x) is called an even function if
[f(−x) = f(x)] for all x in its domain.
Key idea:
Replacing x by −x does not change the value of the function.
2. What is an Odd Function?
A function f(x) is called an odd function if
[f(−x) = −f(x)] for all x in its domain.
Key idea:
Replacing x by −x changes the sign of the function value.
3. Important Observations
- The domain must be symmetric about 0
- A function can be even, odd, or neither
- The zero function is both even and odd
4. Graphical Interpretation
- Even function: symmetric about the y-axis
- Odd function: symmetric about the origin
5. Examples with Solutions
Example 1. Show that [f(x) = x²] is an even function.
Solution:
[f(−x) = (−x)²][ = x² = f(x)]
Conclusion:
[f(−x) = f(x)] ⇒ [f(x)] is even.
Example 2. Show that [f(x) = x³] is an odd function.
Solution:
[f(−x) = (−x)³][ = −x³ = −f(x)]
Conclusion:
[f(−x) = −f(x)] ⇒[f(x)] is odd.
Example 3. Determine whether [f(x) = |x|] is even or odd.
Solution:
[|−x| = |x|]
Conclusion:
[f(−x) = f(x)] ⇒ [f(x)] is even.
Example 4. Determine whether [f(x) = x + 1] is even or odd.
Solution:
[f(−x) = −x + 1]
This is neither equal to [f(x)] nor [−f(x)].
Conclusion:
The function is neither even nor odd.
Example 5. Determine whether [f(x) = 0] is even or odd.
Solution:
[f(−x) = 0 = f(x)] and also [f(−x) = −f(x)]
Conclusion:
The zero function is both even and odd.
6. Conceptual Questions with Solutions
1. What is the basic condition for a function to be even?
A function [f(x)] is even if
[f(−x) = f(x)] for all x in its domain.
This means the function value does not change when [x] is replaced by [−x].
2. What is the basic condition for a function to be odd?
A function [f(x)] is odd if
[f(−x) = −f(x)] for all [x] in its domain.
This means the function value changes sign when x is replaced by [−x].
3. Why must the domain be symmetric about 0 for even or odd functions?
For checking [f(−x)], the value [−x] must also belong to the domain.
Hence, the domain must be symmetric about 0.
4. Can a function be both even and odd?
Yes, the zero function satisfies both
[f(−x) = f(x)] and [f(−x) = −f(x)].
Hence, it is both even and odd.
5. Can a function be neither even nor odd?
Yes. Many functions like [f(x) = x + 1] satisfy neither condition.
Such functions are called neither even nor odd.
6. Is every polynomial an even or odd function?
No.
Polynomials with only even powers are even.
Polynomials with only odd powers are odd.
Mixed powers give neither.
7. Why is [f(x) = x²] an even function?
Because
[(−x)² = x²]
Hence, [f(−x) = f(x)], so the function is even.
8. Why is [f(x) = x³] an odd function?
Because
[(−x)³ = −x³]
Hence, [f(−x) = −f(x)], so the function is odd.
9. Is the modulus function even or odd?
The modulus function satisfies
[|−x| = |x|]
Hence, it is an even function.
10. Is the reciprocal function [f(x) = 1/x] odd?
Yes.
[f(−x) = −1/x][ = −f(x)
Hence, it is an odd function.
11. What is the graph symmetry of an even function?
The graph of an even function is symmetric about the y-axis.
12. What is the graph symmetry of an odd function?
The graph of an odd function is symmetric about the origin.
13. Can an even function have an odd graph shape?
No.
The symmetry of the graph must always match the algebraic property.
14. Is the constant function always even?
Yes.
For [f(x) = c], we have [f(−x) = c][ = f(x).
Hence, it is even.
15. Why is checking [f(−x)] important?
Because replacing [x] by [−x] directly tests whether the function has
symmetry about the y-axis or origin.
7. FAQs / Common Misconceptions
1. If a function looks symmetric, is it automatically even or odd?
No.
You must always verify using algebraic conditions, not just the graph.
2. Is every symmetric graph an even function?
Only if the symmetry is about the y-axis.
3. Can a function be odd if its graph is symmetric about the y-axis?
No.
Odd functions are symmetric about the origin, not the y-axis.
4. Is [f(x) = x² + x] odd because it has x?
No.
Presence of x alone does not decide.
You must check [f(−x)].
5. Can a function with restricted domain be even or odd?
Only if the domain is symmetric about 0.
6. Is the zero function a special case?
Yes.
It is the only function which is both even and odd.
7. Does an odd function always pass through the origin?
Yes.
For odd functions, [f(0) = 0] if [0] belongs to the domain.
8. Is a constant function odd?
No.
It is even unless the constant is zero.
9. Can trigonometric functions be even or odd?
Yes.
For example, cos x is even and sin x is odd.
10. Should we simplify the function before checking?
Yes.
Always simplify first to avoid mistakes.
8. Practice Questions with Step-by-Step Solutions
Question 1. Check whether [f(x) = x⁴ − 2x²] is even or odd.
Step-by-Step Solution:
Find [f(−x)]
[f(−x) ][= (−x)⁴ − 2(−x)² ][= x⁴ − 2x²]
[f(−x) = f(x)]
Conclusion:
The function is even.
Question 2. Check whether [f(x) = x³ − x] is even or odd.
Step-by-Step Solution:
[f(−x) ][= (−x)³ − (−x) ][= −x³ + x]
[f(−x) ][= −(x³ − x) ][= −f(x)]
Conclusion:
The function is odd.
Question 3. Check whether [f(x) = x² + x] is even or odd.
Step-by-Step Solution:
[f(−x) = x² − x]
This is neither [f(x)] nor [−f(x)]
Conclusion:
The function is neither even nor odd.
Question 4. Check whether [f(x) = |x| + x] is even or odd.
Step-by-Step Solution:
[f(−x) = |x| − x]
This is not equal to [f(x)] or [−f(x)]
Conclusion:
The function is neither even nor odd.
Question 5. Check whether [f(x) = 1 / x] is even or odd.
Step-by-Step Solution:
[f(−x) = −1 / x]
[f(−x) = −f(x)]
Conclusion:
The function is odd.
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