Determine the nature of the function for even and odd function

1. Objective

To check whether a given function is:

• Even
• Odd
• Neither even nor odd

2. Standard Method (Step-by-Step)

Step 1. Write the given function as f(x).
Step 2. Replace x by −x to find f(−x).
Step 3. Simplify f(−x).
Step 4. Compare f(−x) with f(x) and −f(x).

• If [f(−x) = f(x)] ⇒ Even
• If [f(−x) = −f(x)] ⇒ Odd
• Otherwise ⇒ Neither

Important Condition:
The domain must be symmetric about 0.

3. Examples with Solutions

Example 1. Determine the nature of [f(x) = x² + 4].

Solution:
[f(−x) ][= (−x)² + 4 = x² + 4 ][= f(x)]

Conclusion:
The function is even.

Example 2. Determine the nature of [f(x) = x³ − 5x].

Solution:
[f(−x) ][= (−x)³ − 5(−x) ][= −x³ + 5x]
[= −(x³ − 5x) ][= −f(x)]

Conclusion:
The function is odd.

Example 3. Determine the nature of [f(x) = x² + x].

Solution:
[f(−x) ][= x² − x]
This is neither equal to [f(x)] nor [−f(x)].

Conclusion:
The function is neither even nor odd.

Example 4. Determine the nature of [f(x) = |x| + x].

Solution:
[f(−x) ][= |x| − x]
This is neither [f(x)] nor [−f(x)].

Conclusion:
The function is neither even nor odd.

Example 5. Determine the nature of [f(x) = 0].

Solution:
[f(−x) ][= 0 = f(x)] and also [= −f(x)]

Conclusion:
The zero function is both even and odd.

4. Practice Questions with Step-by-Step Solutions