# even/odd graph

• Aug 9th 2009, 09:27 AM
live_laugh_luv27
even/odd graph
Are the graphs of

$\sqrt{x}$

$2^x$

$log_2x$

even or odd? I don't see any symmetry with respect to y axis or origin. Thanks!
• Aug 9th 2009, 09:40 AM
Jhevon
Quote:

Originally Posted by live_laugh_luv27
Are the graphs of

$\sqrt{x}$

$2^x$

$log_2x$

even or odd? I don't see any symmetry with respect to y axis or origin. Thanks!

A function $f(x)$ is even if $f(-x) = f(x)$

A function $f(x)$ is odd if $f(-x) = -f(x)$

can you continue?
• Aug 9th 2009, 10:10 AM
live_laugh_luv27
• Aug 9th 2009, 10:15 AM
live_laugh_luv27
I'm still not exactly sure how to set that up, but have another ?...even if a function does not have symmetry in respect to the y-axis, can it still be even? similarly, even if the function does not have symmetry in respect to the origin, can it still be odd?
thanks!
• Aug 9th 2009, 10:17 AM
Plato
Quote:

Originally Posted by live_laugh_luv27

What does any of the above have to do with the basic definitions of odd & even?
$\text{Odd functions are such that }f(-x)=-f(x)$

$\text{Even functions are such that }f(-x)=f(x)$
• Aug 9th 2009, 10:27 AM
Jhevon
Quote:

Originally Posted by live_laugh_luv27
I'm still not exactly sure how to set that up, but have another ?...even if a function does not have symmetry in respect to the y-axis, can it still be even? similarly, even if the function does not have symmetry in respect to the origin, can it still be odd?
thanks!

no, even and odd follow the exact definitions i gave you. you don't even have to look at the graphs.

example, state whether (a) $f(x) = x^2$, (b) $f(x) = \sin x$ and (c) $f(x) = \frac {x^2}{x^3 + 1}$

(a) $f(x) = x^2$ is even, since

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

(b) $f(x) = \sin x$ is odd, since

$f(-x) = \sin (-x) = - \sin x = - f(x)$

(c) $f(x) = \frac {x^2}{x^3 + 1}$ is neither even nor odd, since

$f(-x) = \frac {(-x)^2}{(-x)^3 + 1} = \frac {x^2}{1 - x^3} \ne f(x) \text{ or } -f(x)$

notice that i didn't even draw any graphs. now, try again
• Aug 9th 2009, 10:29 AM
live_laugh_luv27
Quote:

Originally Posted by Plato
What does any of the above have to do with the basic definitions of odd & even?
$\text{Odd functions are such that }f(-x)=-f(x)$

$\text{Even functions are such that }f(-x)=f(x)$

http://www.mathhelpforum.com/math-he...19f85201-1.gif - even

http://www.mathhelpforum.com/math-he...84e8019b-1.gif - odd

http://www.mathhelpforum.com/math-he...069c3f11-1.gif - even
• Aug 9th 2009, 10:31 AM
Jhevon
yes, they are wrong. did you try what i said?
• Aug 9th 2009, 10:33 AM
live_laugh_luv27
http://www.mathhelpforum.com/math-he...19f85201-1.gif = $\sqrt{-x}$ = odd

http://www.mathhelpforum.com/math-he...84e8019b-1.gif = $2^{-x}$ = odd?

http://www.mathhelpforum.com/math-he...069c3f11-1.gif = $\frac{ln(-x )}{ln2}$ ?
• Aug 9th 2009, 10:36 AM
Jhevon
Quote:

Originally Posted by live_laugh_luv27
http://www.mathhelpforum.com/math-he...19f85201-1.gif = $\sqrt{-x}$ = odd

?! are you telling me that, for example, $\sqrt 2 = \sqrt {-2}$? and if so, does that comply with the definition of an odd function as defined in my and Plato's posts?

Quote:

http://www.mathhelpforum.com/math-he...84e8019b-1.gif = $2^{-x}$ = odd?
interesting. say $x = 1$, is it true that $2^1 = 2^{-1}$ or $-2^1 = 2^{-1}$ ?

Quote:

http://www.mathhelpforum.com/math-he...069c3f11-1.gif = $\frac{ln(-x )}{ln2}$ ?
tell me, what's $\log_2 (-5)$, say
• Aug 9th 2009, 10:52 AM
live_laugh_luv27
http://www.mathhelpforum.com/math-he...19f85201-1.gif = http://www.mathhelpforum.com/math-he...a3ae9cbf-1.gif = $-\sqrt{x}$ = $-f(x)$ ?

http://www.mathhelpforum.com/math-he...2950423e-1.gif is not true

http://www.mathhelpforum.com/math-he...f1a02f81-1.gif is $\frac{ln(-5)}{ln(2)}$ ?
• Aug 9th 2009, 10:57 AM
Jhevon
Quote:

Originally Posted by live_laugh_luv27
http://www.mathhelpforum.com/math-he...19f85201-1.gif = http://www.mathhelpforum.com/math-he...a3ae9cbf-1.gif = $-\sqrt{x}$ = $-f(x)$ ?

http://www.mathhelpforum.com/math-he...2950423e-1.gif is not true

http://www.mathhelpforum.com/math-he...f1a02f81-1.gif is $\frac{ln(-5)}{ln(2)}$ ?

ok, as far as real numbers are concerned, the square root function and the logarithm function are not defined for negative real numbers.... you should know this. please look this up and make sure you get it

and clearly $2^1 \ne 2^{-1}$ nor does $-2^1 = 2^{-1}$ so that $f(x) = 2^x$ is neither even nor odd.