EVEN FUNCTION

Even Function

Definition Of Even Function

The function f(x) is said to be 'even' if and only if f(x) is a real-valued function of a real variable x, and f(-x) = f(x).

More About Even Function

Sum of an even and an odd function is neither even nor odd.
Sum of two even functions is even, and any constant multiple of an even function is even.
The product of two even functions is even.

Video Examples: Even Odd Functions & Symmetry

Example of Even Function

The function f(x) = - 3x² + 4 is an even function as:
f(- x) = - 3(- x)² + 4
= - 3(x)² + 4 = f(x)
So, f(- x) = f(x)

Solved Example on Even Function

Ques: Find whether the function f(x) = x⁸ + x² - 7 is an even function, or an odd function, or neither.

Choices:

A. neither even nor odd
B. data insufficient
C. odd function
D. even function
Correct Answer: D

Solution:

Step 1: f(x) = x⁸ + x² - 7
Step 2: f(- x) = (- x)⁸ + (- x)² - 7 = x⁸ + x² - 7
Step 3: Since both f(x)and f(- x)are equal, the function is an even function.

Quick Summary

Even functions are symmetric with respect to the y-axis.
The sum of two even functions is even.
The product of two even functions is even.

\[ f(-x) = f(x) \]

🍎 Teacher Insights

Use graphical representations to illustrate symmetry. Emphasize the algebraic verification using f(-x).

🎓 Prerequisites

Algebra
Basic function notation

Check Your Knowledge

Q1: Which of the following functions is even?

A. f(x) = x B. f(x) = x^2 C. f(x) = x^3 D. f(x) = x + 1

Q2: If f(x) is an even function and f(2) = 5, what is f(-2)?

A. -5 B. 2 C. 5 D. -2

Frequently Asked Questions

Q: Is f(x) = x^3 an even function?
A: No, because f(-x) = (-x)^3 = -x^3 = -f(x), so it's an odd function.

Q: Is f(x) = x^2 + 1 an even function?
A: Yes, because f(-x) = (-x)^2 + 1 = x^2 + 1 = f(x).