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

[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.

 

[CompuChip]

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.

 

[HallsofIvy]

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.

Why was this titled "decibal"?