Expert: Paul Klarreich - 10/20/2006

Question
How do I find a even function?

For example: f(x)=4x-2
f(x)=3-x^2 and f(x)=x/x+1


Answer
Questioner:  Francis
Category:  Calculus
 
Subject:  even function
Question:  How do I find a even function?

For example: f(x)=4x-2
f(x)=3-x^2 and f(x)=x/x+1
........................................
Hi, Francis,

Question:  How do I find a even function?
Answer: Take a right at the stop sign, go three blocks, and...  oops, that's another question.

The definition for an even function is:

f(x) is even if  f(-x) = f(x).  In practice, you substitute (-x), with the parentheses and all, and simplify the result.  If it comes out identical to f(x), f is even.

So, for your first example:

f(x) = 4x - 2.
f(-x) = 4(-x) - 2 = - 4x - 2.
That's not the same as f(x); this f(x) is not even.
.................................
Then:
f(x) = 3 - x^2
f(-x) = 3 - (-x)^2 = 3 - (x^2) = 3 - x^2.
That is the same as f(x); this function is even.
.................................
Then:
        x    
f(x) = -----
      x - 1
         (-x)      - x       x
f(-x) = -------- = ------ = -----
       (-x) - 1   -x - 1   x + 1

That's not the same as  f(x), this function is not even.
.......................................

Notes:
A. If  f(-x) = - f(x), then f(x) is an ODD function.
B. Most functions are neither even nor odd.
C. An even function has a graph with left-right symmetry.
D. An odd function has a graph with origin symmetry.
E. Any function can be decomposed into an odd component and an even component. [Method on request.]