Math Help - Even, Odd, or Neither

1. 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??

2. 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

3. can someone show me an example like question a) so I can have a better understanding thanks

4. 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 59th post!!!!

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