CAT Even Functions and Odd Functions - Practice Questions & MCQ

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.