NIMCET 2022 — Mathematics PYQ
NIMCET | Mathematics | 2022The function f(x)=log(x+x2+1) is
Choose the correct answer:
- A.
An even function
- B.
An odd function
(Correct Answer) - C.
A periodic function
- D.
Neither an even nor an odd function
An odd function
Explanation
To determine if a function is even or odd, we replace x with −x and examine the relationship between f(−x) and f(x).
1. Recall the Definitions:
-
A function is even if f(−x)=f(x).
-
A function is odd if f(−x)=−f(x).
2. Substitute −x into the function:
Given: f(x)=log(x+x2+1)
3. Use Rationalization:
To compare f(−x) with f(x), we multiply and divide the argument by its conjugate (x2+1+x):
Applying the identity (a−b)(a+b)=a2−b2:
4. Apply Logarithmic Properties:
Using the property log(a1)=−log(a):
Conclusion:
Since f(−x)=−f(x), the function is an odd function.
Correct Option:
B) An odd function
Explanation
To determine if a function is even or odd, we replace x with −x and examine the relationship between f(−x) and f(x).
1. Recall the Definitions:
-
A function is even if f(−x)=f(x).
-
A function is odd if f(−x)=−f(x).
2. Substitute −x into the function:
Given: f(x)=log(x+x2+1)
3. Use Rationalization:
To compare f(−x) with f(x), we multiply and divide the argument by its conjugate (x2+1+x):
Applying the identity (a−b)(a+b)=a2−b2:
4. Apply Logarithmic Properties:
Using the property log(a1)=−log(a):
Conclusion:
Since f(−x)=−f(x), the function is an odd function.
Correct Option:
B) An odd function