# Even, Odd, or Neither

• Dec 31st 2007, 05:21 PM
johett
Even, Odd, or Neither
Determine whether each of the following functions are even, odd, or neither. Please state the symmetry.

a)f(x)=1 / (x^2+1) b)g(x)=x^2 / (1+x^4) c)h(x)= (2x^4) + (3x^2)
d)f(x)={1 / (x^3+x)}^5 e)g(x)=x+(1/x) f)h(x)=x-(x^2)
g)f(x)=2^x h)g(x)=(logx^2) / (log3x^4)

For each of the above functions from a) through h) how do you determine if it is even,odd,or neither??
• Dec 31st 2007, 05:29 PM
Jhevon
Quote:

Originally Posted by johett
Determine whether each of the following functions are even, odd, or neither. Please state the symmetry.

a)f(x)=1 / (x^2+1) b)g(x)=x^2 / (1+x^4) c)h(x)= (2x^4) + (3x^2)
d)f(x)={1 / (x^3+x)}^5 e)g(x)=x+(1/x) f)h(x)=x-(x^2)
g)f(x)=2^x h)g(x)=(logx^2) / (log3x^4)

For each of the above functions from a) through h) how do you determine if it is even,odd,or neither??

a function is even if f(x) = f(-x). that is, if you replace x with -x and simplify, you get exactly the original function

a function is odd if f(-x) = -f(x). that is, if you replace x with -x and simplify, you get the negative of the original function

see the "Ways to test for symmetry" section here on how to test for other kinds of symmetry. i don't know what kinds you are expected to look for.

note: symmetry about the y-axis is the same as being even, and symmetry about the origin is the same as being odd
• Dec 31st 2007, 05:35 PM
johett
can someone show me an example like question a) so I can have a better understanding :) thanks
• Dec 31st 2007, 05:44 PM
Jhevon
Quote:

Originally Posted by johett
can someone show me an example like question a) so I can have a better understanding :) thanks

i'll give you other examples. then try your questions and post your solutions and we'll tell you if you applied the rules correctly.

example 1:

the function $f(x) = x^2 + 1$ is even, since replacing x with -x we get:

$f(-x) = (-x)^2 + 1 = x^2 + 1 = f(x)$

since $f(-x) = f(x)$, the function is even

example 2:

the function $f(x) = x^3$ is odd, since if we replace x with -x we get:

$f(-x) = (-x)^3 = -x^3 = -f(x)$

since $f(-x) = -f(x)$, the function is odd

example 3:

the function $f(x) = x^3 + 3$ is neither even nor odd, since if we replace x with -x we get:

$f(-x) = (-x)^3 + 3 = -x^3 + 3$

clearly $f(-x) \ne f(x)$ and $f(-x) \ne -f(x)$. so the function is neither even nor odd

This is my 59:):)th post!!!!
• Dec 31st 2007, 05:48 PM
topsquark
Quote:

Originally Posted by johett
can someone show me an example like question a) so I can have a better understanding :) thanks

Quote:

Originally Posted by johett
Determine whether each of the following functions are even, odd, or neither. Please state the symmetry.

a)f(x)=1 / (x^2+1)
d)f(x)={1 / (x^3+x)}^5
f)h(x)=x-(x^2)

For each of the above functions from a) through h) how do you determine if it is even,odd,or neither??

a)
$f(x) = \frac{1}{x^2 + 1}$

$f(-x) = \frac{1}{(-x)^2 + 1}$

$f(-x) = \frac{1}{x^2 + 1} = f(x)$
so this function is even.

d)
$f(x) = \left ( \frac{1}{x^3 + x} \right ) ^5$

$f(-x) = \left ( \frac{1}{(-x)^3 + (-x)} \right ) ^5$

$f(-x) = \left ( \frac{1}{-x^3 - x} \right ) ^5$

$f(-x) = \left ( \frac{-1}{x^3 + x} \right ) ^5$

$f(-x) = (-1)^5 \left ( \frac{1}{x^3 + x} \right ) ^5$

$f(-x) = - \left ( \frac{1}{x^3 + x} \right ) ^5 = -f(x)$

so this function is odd.

f)
$h(x) = x - x^2$

$h(-x) = (-x) - (-x)^2$

$h(-x) = -x + x^2$

which is equal to neither h(x) nor -h(x). So this function is neither even, nor odd.

-Dan