In simple terms, how do you find the domain and range of a function? 

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Let's start from the domain. Usually this word is used in the sense "all values of `x` which are possible to use for the given formula". There are some things which is impossible: division by zero, finding square root of the negative number, finding logarithm of non-positive number and so on.

So when we see a formula, we have to analyse whether it contains division, roots and so on. Example:


There is `x` in the denominator, so `x!=0.` Also, there is a square root, so `x+2gt=0` or `xgt=-2.` The resulting domain is `[-2,0) uu (0,+oo).`

Sometimes finding the domain may be difficult, but the idea is as above. Try the function `g(x)=sqrt(2-sqrt(3-x))` (isn't very difficult).


The range of a function is the set of its values. To decide whether some `y` is in range of `f(x)` we have to consider the equation `f(x)=y` for `x.` If at least one solution exists, then `y` is in the range.

For example, the range of a linear function `f(x) =ax+b` is the set of all real numbers if `a!=0,` and the set of the only one element `{b}` if `a=0.` Another example: you probably know that the function `g(x)=sin(x)` takes all values in `[-1,1]` but no values outside; this means that the range of `sin(x)` is `[-1,1].`

Finding a range may also be difficult.


Note that sometimes it is useful to restrict a domain "manually". For example, one may consider the function `h(x)=x^2+1` on the segment `[1,2]` only. Then its domain is `[1,2]` and the range is `[2,5]`, while for the "unrestricted" function the domain is all real numbers and the range is `[1,+oo).`


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