Physics and Mathematics
Sum, Difference, Product and Quotient of Even and Odd Functions
1. Key Idea
Let [f(x)] and [g(x)] be two functions defined on a domain symmetric about [0].
We examine the nature of
[f(x) ± g(x)] using the definitions of even and odd functions.
2. Rules for Sum and Difference
| [f(x)] | [g(x)] | [f(x) + g(x)] | [f(x) − g(x)] |
|---|---|---|---|
| Even | Even | Even | Even |
| Odd | Odd | Odd | Odd |
| Even | Odd | Neither | Neither |
Important Note:
- Sum or difference of even and odd is generally neither even nor odd.
3. Product and Quotient of Even and Odd Functions
Key Idea
Let [f(x)] and [g(x)] be two functions defined on a domain symmetric about 0.
We examine the nature of
[f(x)g(x)] and [f(x)/g(x)].
4. Rules for Product and Quotient
| [f(x)] | [g(x)] | [f(x)g(x)] | [f(x)/g(x)] |
|---|---|---|---|
| Even | Even | Even | Even |
| Odd | Odd | Even | Even |
| Even | Odd | Odd | Odd |
Important Note:
- Quotient exists only when [g(x) ≠ 0].
5. Examples with Solutions
Example 1. If [f(x) = x²] and [g(x) = x⁴], find the nature of [f(x) + g(x)].
Solution:
Both functions are even.
Sum of even functions is even.
Conclusion:
[f(x) + g(x)] is even.
Example 2. If [f(x) = x³] and [g(x) = x], find the nature of [f(x) − g(x)].
Solution:
Both functions are odd.
Difference of odd functions is odd.
Conclusion:
[f(x) − g(x)] is odd.
Example 3. If [f(x) = x²] and [g(x) = x], find the nature of [f(x)g(x)].
Solution:
[x²] is even, x is odd.
Product of even and odd is odd.
Conclusion:
[f(x)g(x)] is odd.
Example 4. If [f(x) = x³] and [g(x) = x], find the nature of [f(x)/g(x)].
Solution:
Both functions are odd.
Quotient of odd and odd is even.
Conclusion:
[f(x)/g(x)] is even.
Example 5. If [f(x)] is even and [g(x)] is odd, find the nature of [f(x) − g(x)].
Solution:
Difference of even and odd functions is neither even nor odd.
Conclusion:
[f(x) − g(x)] is neither.
6. Conceptual Questions with Solutions
1. Why is the sum of two even functions always even?
If f(x) and g(x) are even, then
[f(−x) = f(x)] and [g(−x) = g(x)]
So,
[f(−x) + g(−x) = f(x) + g(x)]
Hence, the sum is even.
2. Why is the difference of two even functions also even?
Since both functions satisfy
[f(−x) = f(x)] and [g(−x) = g(x)],
we get
[f(−x) − g(−x) = f(x) − g(x)]
Hence, the difference is even.
3. Why is the sum of two odd functions odd?
For odd functions,
[f(−x) = −f(x)] and [g(−x) = −g(x)]
Adding,
[f(−x) + g(−x) = −(f(x) + g(x))]
Hence, the sum is odd.
4. Why is the difference of two odd functions odd?
For odd functions,
[f(−x) − g(−x) = −f(x) + g(x) = −(f(x) − g(x))]
Hence, the difference is odd.
5. Why is the sum of an even and an odd function neither even nor odd?
For f(x) even and g(x) odd,
[f(−x) = f(x) − g(x)]
This is neither equal to f(x) + g(x) nor its negative.
Hence, the sum is neither.
6. Why is the product of two even functions even?
If both are even,
[f(−x)g(−x) = f(x)g(x)]
Hence, the product is even.
7. Why is the product of two odd functions even?
For odd functions,
[f(−x)g(−x) = (−f(x))(−g(x)) = f(x)g(x)]
Hence, the product is even.
8. Why is the product of an even and an odd function odd?
If f(x) is even and g(x) is odd,
[f(−x)g(−x) = f(x)(−g(x)) = −f(x)g(x)]
Hence, the product is odd.
9. Why is the quotient of two even functions even?
If both are even and g(x) ≠ 0,
[f(−x)/g(−x) = f(x)/g(x)]
Hence, the quotient is even.
10. Why is the quotient of two odd functions even?
For odd functions,
[f(−x)/g(−x) = (−f(x))/(−g(x)) = f(x)/g(x)]
Hence, the quotient is even.
11. Why is the quotient of even and odd functions odd?
If numerator is even and denominator is odd,
[f(−x)/g(−x) = f(x)/(−g(x)) = −f(x)/g(x)]
Hence, the quotient is odd.
12. Why must the domain be symmetric for these rules?
To evaluate f(−x) and g(−x), −x must belong to the domain.
Without symmetry, even–odd classification is not valid.
13. Can sum or product rules be applied without checking domain?
No.
The domain condition is essential before applying any rule.
14. Does the zero function affect these rules?
Yes.
The zero function is both even and odd, so results must be interpreted carefully.
15. Why are these rules important in calculus?
They simplify
integration limits,
graph analysis, and
function classification.
7. FAQs / Common Misconceptions
1. Is the sum of even and odd always odd?
No.
It is generally neither even nor odd.
2. Can the product of two odd functions be odd?
No.
Odd × Odd is always even.
3. Is the quotient rule always applicable?
No.
The denominator must satisfy g(x) ≠ 0.
4. Can we decide nature just by looking at formulas?
No.
You must always verify using f(−x).
5. Does even + even ever become odd?
No.
Even + Even is always even.
6. Is odd − odd always even?
No.
Odd − Odd is odd.
7. Can restricted domains change the result?
Yes.
Without symmetry about 0, rules may fail.
8. Are these rules valid for trigonometric functions?
Yes, provided their even–odd nature is known.
9. Is constant × odd always odd?
Yes, if the constant is non-zero.
10. Should we memorize or understand these rules?
They should be understood, not just memorized, to avoid mistakes.
8. Practice Questions with Step-by-Step Solutions
Practice Question 1. If [f(x) = x²] and [g(x) = x⁶], find the nature of [f(x) + g(x)].
Step-by-Step Solution:
[x²] is even.
[x⁶] is even.
Sum of even functions is even.
Conclusion:
[f(x) + g(x)] is even.
Practice Question 2. If [f(x) = x³] and [g(x) = x⁵], find the nature of [f(x) − g(x)].
Step-by-Step Solution:
Both functions are odd.
Difference of odd functions is odd.
Conclusion:
[f(x) − g(x)] is odd.
Practice Question 3. If [f(x)] is even and [g(x)] is odd, find the nature of [f(x)g(x)].
Step-by-Step Solution:
Even × Odd = odd.
Conclusion:
[f(x)g(x)] is odd.
Practice Question 4. If [f(x)] is odd and [g(x)] is odd, find the nature of [f(x)g(x)].
Step-by-Step Solution:
Odd × Odd = even.
Conclusion:
[f(x)g(x)] is even.
Practice Question 5. If [f(x)] is even and [g(x)] is odd, find the nature of [f(x)/g(x)].
Step-by-Step Solution:
Even ÷ Odd = odd (where defined).
Conclusion:
[f(x)/g(x)] is odd.
Practice Question 6. If [f(x) = x²] and [g(x) = x], find the nature of [f(x) + g(x)].
Step-by-Step Solution:
[x²] is even.
[x] is odd.
Sum of even and odd is neither.
Conclusion:
[f(x) + g(x)] is neither.
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