TheMathPage

Topics in

P R E C A L C U L U S

Table of Contents | Home

16

SYMMETRY

Test for symmetry:  Even and odd functions

LET THIS BE THE RIGHT-HAND SIDE of the graph of a function:

Symmetry

We will now draw the left-hand side -- so that the graph will be symmetrical with respect to the y-axis:

Symmetry

In this case,

f(−x) = f(x).

The height of the curve at −x is equal to the height of the curve at x -- for every x in the domain of f.

Again, let this be the right-hand side:

Symmetry

We will now draw the left-hand side -- so that the graph will be symmetrical with respect to the origin:

Symmetry

Every point on the right-hand side is reflected through the origin.  In this case,

f(−x) = −f(x).

The value of f at −x is the negative of the value at x.

(A reflection through the origin is equivalent to a reflection about the y-axis, followed by a reflection about the x-axis.)

Test for symmetry: Even and odd functions

Symmetry, then, depends on the behavior of f(x) on the other side of the y-axis -- at minus-x :   f(−x).

Here is the test:

If f(−x) = f(x),

then the graph of f(x) is symmetrical with respect to the y-axis.

To see the answer, pass your mouse over the colored area.
To cover the answer again, click "Refresh" ("Reload").

If f(−x) = −f(x),

then the graph of f(x) is symmetrical with respect to the origin.

A function symmetrical with respect to the y-axis is called an even function.

A function that is symmetrical with respect to the origin  is called an odd function.

Example 1.   Test this function for symmetry:

f(x) = x4 + x2 + 3

 Solution.   We must look at f(−x):

f(−x)  =  (−x)4 + (−x)2 + 3
 
   =  x4 + x2 + 3
 
   =  f(x).

Since  f(−x) = f(x), this function is symmetrical with respect to the y-axis. It is an even function.

Example 2.   Test this function for symmetry:

f(x) = x5 + x3 + x

 Solution.   Again, we must look at f(−x):

f(−x)  =  (−x)5 + (−x)3 + (−x)
 
   =  x5x3x
 
   =  −(x5 + x3 + x)
 
   =  f(x).

Since  f(−x) = −f(x), this function is symmetrical with respect to the origin. It is an odd function.

Problem.   Test each of the following for symmetry.  Is f(x) even, odd, or neither?

a)  f(x) = x3 + x2 + x + 1

Answer.   Neither, because f(−x) ≠ f(x) , and f(−x) ≠ −f(x).

b)  f(x) = 2x3 − 4x

Answer.   f(x) is odd -- it is symmetrical with respect to the origin -- because f(−x) = −f(x).

c)  f(x) = 7x2 − 11

Answer.   f(x) is even -- it is symmetrical with respect to the y-axis -- because f(−x) = f(x).

Note:  A polynomial will be an even function when all the exponents are even.

A polynomial will be an odd function when all the exponents are odd.

But there are even and odd functions that are not polynomials. In trigonometry,  y = sin x is an odd function, while y = cos x is even.

Therefore, the issue is the test of f(−x).

End of the lessson

Next Topic:  Translations


Table of Contents | Home


Please make a donation to keep TheMathPage online.
Even $1 will help.


Copyright © 2014 Lawrence Spector

Questions or comments?

E-mail:  themathpage@nyc.rr.com