# Square Root (x(5-x)): Natural Domain & Proving Algebraically

• imdapolak
The natural domain is the domain that makes sense for the function, in this case all non-negative real numbers.f

#### imdapolak

1. What is natural Domain of Square root(x(5-x))

2.

3. I know x can not make the square root equal zero, but I am not sure how to prove this algebraically. Any help would be appreciated, also what's the difference between natural doman, and Domain? My book doesn't explain this at all.

Making the square root zero is not a problem, making the argument (what's inside it) negative is. You can calculate when that is by first looking when it will be precisely zero though.

As for the second question: the domain of the function needs to be given. For example, you can define the function on [1, 2] only. By natural domain we mean the largest domain on which defining the function makes sense, for example [0, $\infty$) for sqrt(x) and $x \neq 0$ for 1/x.

1. What is natural Domain of Square root(x(5-x))

2.

3. I know x can not make the square root equal zero,

No, you don't know that! The argument of a square root certainly can be 0, it just can't be negative.
That is x(5-x) must be greater than or equal to 0. Now a product of numbers will be larger than or equal to 0, if and only if both factors are positive or 0, $x\ge 0$ and $5- x\ge 0$, or both factors are negative, $x\le 0$ and $5- x\ge 0$.

but I am not sure how to prove this algebraically. Any help would be appreciated, also what's the difference between natural doman, and Domain? My book doesn't explain this at all.
A function consists of (a) a domain- the possible values of x and (b)a rule for finding the y value corresponding to each x value. Often we are given only the function ("rule") y= f(x). In that case, the presumed domain, the "natural domain" is the set of all x values for which the equation can be calculated. For example, if $f(x)= \sqrt{x}$, then the "natural domain" is the set of all non-negative real numbers. But we are also free to state the domain as a subset of that. For example, I can define the function F(x) by the rule $f(x)= \sqrt{x}$ with the domain restricted to "all x larger than 1". That is now a different function than f and has domain "all real numbers larger than 1". Or I could define g(x) by $g(x)= \sqrt{x}$ with domain "all positive integers" or G(x) by $G(x)= \sqrt{x}$ with domain "all positive rational numbers". Those are 4 different functions, with the same rule but different domains. The "natural domain" for the rule $y= \sqrt{x}$ is the set of all non-negative real numbers.