What is the domain of this function? (I am so confused)

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November 17th 2006, 01:59 PM
Val21

What is the domain of this function? (I am so confused)

Hello.

I would like to ask what is the domain of the following fucnction.

y=(5x^x)^(x+1)

Our teachers tell us that the domain of a power function (A^X where A is constant and X is a variable) is that the base must be positive and must differ from 1,so in the above example, it is logically that the solution of the following system of inequalities defines the domain:

5x>0 and
5x<>1 and
5x^x>0 and
5x^x<>1

Am I right? And I must note that in our textbook the power function's domain(sorry if I misuse mathematical argo) A^X is that the base A must be positive but it does not say that it must differ from 1. Ohhhh! Please clarify this very obscure situation!
November 17th 2006, 04:48 PM
topsquark

Quote:

Originally Posted by Val21

Hello.

I would like to ask what is the domain of the following fucnction.

y=(5x^x)^(x+1)

Our teachers tell us that the domain of a power function (A^X where A is constant and X is a variable) is that the base must be positive and must differ from 1,so in the above example, it is logically that the solution of the following system of inequalities defines the domain:

5x>0 and
5x<>1 and
5x^x>0 and
5x^x<>1

Am I right? And I must note that in our textbook the power function's domain(sorry if I misuse mathematical argo) A^X is that the base A must be positive but it does not say that it must differ from 1. Ohhhh! Please clarify this very obscure situation!

$y=\left ( 5x^x \right )^{x+1} = 5^{x+1} \cdot x^{x(x+1)}$

Looking at these terms individually, we see that we have no restrictions on the function due to the first factor. The second factor gives us that $x \geq 0$ , since the base must be non-negative, but note that if x = 0 the second factor is $0^0$ , which is undefined. So the domain is x > 0.

-Dan
November 17th 2006, 11:45 PM
Val21

Must a base differ from 1?

And what if we take X for 1 so we would get number 1 in the base ,which is prohibited. So probably the domain must be x>0 and x<>1 ?

Thanks for replying
November 17th 2006, 11:54 PM
CaptainBlack

Quote:

Originally Posted by Val21

And what if we take X for 1 so we would get number 1 in the base ,which is prohibited. So probably the domain must be x>0 and x<>1 ?

Thanks for replying

$x$ can also be any integer, so it looks to me as though the domain is:

$<br /> \left[ \mathbb{R}_+ \cup \mathbb{Z} \right] \backslash \{0\} <br />$

The union of all positive reals, and non-zero integers.

On second thoughts it looks as though this is also defined when $x$ is a rational of
the form $-\frac{2n}{2m-1},\ n,\ m \in \mathbb{Z}_+$ :eek: (negative rationals which in lowest form have an even numerator)

So my latest bet is:

$<br /> \left[ \mathbb{R}_+ \cup \left( \mathbb{Z}\backslash \{0\} \right) \cup A \right]<br />$

where $A=\{a: a=-2n/(2m-1),\ n,\ m \in \mathbb{Z}_+ \}$ .

But somehow I don't think your teacher is looking for an answer of this nature.

RonL
November 17th 2006, 11:57 PM
CaptainBlack

Quote:

Originally Posted by Val21

And what if we take X for 1 so we would get number 1 in the base ,which is prohibited. So probably the domain must be x>0 and x<>1 ?

Thanks for replying

I don't see why $1$ should be prohibited in the base, $0$ yes as $0^0$ is undefined, but not $1$ .

When $x=1$ , we have:

$y=(5x^x)^{x+1}=(5 \times 1^1)^2=5^2=25$

RonL