# What type of function is this?

• Sep 13th 2009, 07:54 AM
mwok
What type of function is this?
y = -x^(1/3)

What type of function is this? Simplified it is y = i*cube root(x)

I'm trying to find the symmetry using the formula x = -b/2a.
• Sep 13th 2009, 08:11 AM
skeeter
Quote:

Originally Posted by mwok
y = -x^(1/3)

What type of function is this? Simplified it is y = i*cube root(x)

I'm trying to find the symmetry using the formula x = -b/2a.

the function you cite is actually $f(x) = -\sqrt[3]{x}$

did you mean $y = (-x)^{\frac{1}{3}}$ ?

if so, this is the same as $f(x) = -\sqrt[3]{x}$

note that $(-x)^{\frac{1}{3}} = (-1)^{\frac{1}{3}} \cdot x^{\frac{1}{3}} = -1 \cdot x^{\frac{1}{3}}$

no i is involved ... $(-1)^{\frac{1}{2}} = i$

$f(x) = -\sqrt[3]{x}$ is an odd function symmetric to the origin.
• Sep 13th 2009, 08:19 AM
mwok
But how did you find the symmetry and odd/even?

I know that an even/odd power corresponds to a even/odd function, however...the power in this case is a fraction. Without graphing, how can I determined this?

BTW, what graphing tool did you use to generate that graph?

Thanks.
• Sep 13th 2009, 08:26 AM
skeeter
Quote:

Originally Posted by mwok
But how did you find the symmetry and odd/even?

I know that an even/odd power corresponds to a even/odd function, however...the power in this case is a fraction. Without graphing, how can I determined this?

BTW, what graphing tool did you use to generate that graph?

Thanks.

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

$f(x) = \sqrt[3]{-x} = -\sqrt[3]{x}$

$f(-x) = \sqrt[3]{-(-x)} = \sqrt[3]{x}$

$f(-x) = \sqrt[3]{x}$ is the opposite of $f(x) = -\sqrt[3]{x}$ , therefore $f(x)$ is odd.

free graph program ...

Graph
• Sep 13th 2009, 08:30 AM
Plato
Quote:

Originally Posted by mwok
But how did you find the symmetry and odd/even?
I know that an even/odd power corresponds to a even/odd function, however...the power in this case is a fraction. Without graphing, how can I determined this?

The names odd & even most likely came from exponents.
But exponents really do not determine much.
For example: $\cos(x)$ is even and $\sin(x)$ is odd.

Learn the actual definitions: If $f(x)=f(-x)$ then $f$ is even.
If $-g(x)=g(-x)$ then $g$ is odd.
• Sep 13th 2009, 08:35 AM
mwok
Thanks, how do you find the symmetry? I cannot use x = -b/2a because that is for polynomials and the function above is not a polynomial.
• Sep 13th 2009, 08:54 AM
Plato
Quote:

Originally Posted by mwok
Thanks, how do you find the symmetry?

First do as I suggested: Learn the basic definitions.
Then it is simple: odd functions are symmetric about the origin; even functions are symmetric about the y-axis.
• Sep 13th 2009, 10:36 AM
mwok
So, the answer is simply that the symmetry is about the origin?
• Sep 13th 2009, 10:40 AM
Plato
Quote:

Originally Posted by mwok
So, the answer is simply that the symmetry is about the origin?

Well of course, if it is an odd function.
$f(x) = - x^{\frac{1}
{3}} \; \Rightarrow \;f( - x) = - \left( { - x} \right)^{\frac{1}
{3}} = x^{\frac{1}
{3}} = - f(x)$
so it is odd.