Function and limits

[Aftab hassan]

Define domain and range?

 

[lev888]

Your text book should have the definitions.

 

[Aftab hassan]

There is problem for me to understand example .

 

[Dr.Peterson]

Please be specific. What definitions and examples were you given, and what do you not understand about them?

 

[Aftab hassan]

Now what will be domain and range of this question which is mention in image.

 

[pka]

Define domain and range?

As an 'old-timer' I will tell you the everything about those two concepts depends upon your particular text-material.
Here is the way I write about it:
The statement that \(\displaystyle f: A\to B\) means that
1) \(\displaystyle f\subset A\times B\)
2) \(\displaystyle (\forall x\in A)(\exists y\in B)[(x,y)\in f]\)
3) No two pairs in \(\displaystyle f\) has the same first term.
We write that if \(\displaystyle (a,b)\in f\) then \(\displaystyle f(a)=b\).
The domain of \(\displaystyle f\) is \(\displaystyle A\) and the final set(range/codomain) is the set of second terms of \(\displaystyle f\).

 

[Aftab hassan]

As an 'old-timer' I will tell you the everything about those two concepts depends upon your particular text-material.
Here is the way I write about it:
The statement that \(\displaystyle f: A\to B\) means that
1) \(\displaystyle f\subset A\times B\)
2) \(\displaystyle (\forall x\in A)(\exists y\in B)[(x,y)\in f]\)
3) No two pairs in \(\displaystyle f\) has the same first term.
We write that if \(\displaystyle (a,b)\in f\) then \(\displaystyle f(a)=b\).
The domain of \(\displaystyle f\) is \(\displaystyle A\) and the final set(range/codomain) is the set of second terms of \(\displaystyle f\).

I uploaded the image above please describe it.

 

[HallsofIvy]

pka gave the most general definition of "function". A function consists of a set of ordered pairs, (x, y). The set of first members, the set of all "x" values, is the "domain" and the set of second members, the set of all "y" values, is the "range". Then there is some "rule", which might be an arithmetic formula but doesn't have to be, that tells what unique "y" is paired with each "x" value. For some functions, we can use a kind of "shorthand" where we give just the formula that connects "x" and "y" with the understanding that the "domain" is the set of all real numbers for which we can apply the formula. For example, if we give a function as simply "\(\displaystyle f(x)=\frac{1}{x}\)" or "\(\displaystyle y= \frac{1}{x}\) the convention is that the domain is the set of all real numbers except 0 since "\(\displaystyle \frac{1}{0}\)" does not exist.

In the example you give, the "formula" is \(\displaystyle f(x)= \sqrt{9- x^2}\). The domain is the set of all real numbers, -3 to 3 inclusive because if x is larger than 3 or less than -3 \(\displaystyle 9- x^2\) will be negative and the square root will not be a real number. Determining the range of a function is typically harder since it is not immediately "given". I note that if x= 3 or -3, y= 0. And if x= 0, y= 3. Further, I can see that every value of x between -3 and 3 gives a value of y between 0 and 3. The range is the set of all real numbers from 0 to 3 inclusive.