Calculating Domain and Range of a function

I have knowledge about the domail and range of a function but sometimes it is confusing for me to calculate the domain and range of a function(without using any software)

for example
1.   f(x) = (x^2 - 3x + 2)/(x + 1)   , x <> -1
Dom f = R - [-1]   and
Range f = R - (-5 -2*sqrt(6),-5+2*sqrt(6))

2.  g(x) = 6x + 7 when x <= -2
           = 4 - 3x when -2 < x

Dom g = Real Set
Range g = (-ve infinity, 10)

The above are two sample question. i want to understand how calculate the domain and especially range of a function.

Is there any software to just verify the answer.

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The two terms come from discrete mathematics, indeed from maps. If x maps to y there is an association for values of x to values of y. One would write  m as a map :-

      m = [x->y]

Ie, a set of mappings. Then the domain of the map is the set of x values and the range the set of y values :-

        Dom(m) = [x],  Range(m) = [y]

So applied to "continuous functions" the domain of the function is the values for which the function exists, the range the values which the function can take.

For f(x) = x, the domian and range are the same, [-oo,+oo]

In f(x) = x² the domain is still [-oo,+oo] (since for all these possible values the function has a result), but the range is [0,+oo] since no negative number may be a square.

In f(x)=sqrt(1-x²) the domain and range are [-1,+1], ie: confined within the unit circle centered on the origin.

So, the best way to calculate domain and range is to draw a graph. This will show (y axis) which range values may be taken for which domian values (x axis).

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For real functions the domain is usually all the reals unless some are excluded explicitly. (In your ex 1 x<> -1.) You cannot calculate the domain. It must be obtained by inspection of the function.
As Big Rat says a graph will help with the range. But if the function becomes large or has a chopped up domain, a little thought helps. Beware of squares and square roots. If you differentiate the function twice and set to zero you will locate all extreme values.  That will give the range for a continuous function. T
There is no software for generally calculating the range but any graphing software will help.

For practice consider
Y = 2*x^2
R = real set
D = o to +infinity
Note the second derivative is zero at x = 0 where y = 0

Y = 2*x^2   for –2 <= x >= 2
Y = -20        for –2 > x < 2
R = real set
D = 0 to infinity plus –20
Note that the second derivative method works here too (even though it is not continuous)
hyynesAuthor Commented:
how about my example especially 2nd example???
For 2) the x in g(x) takes all values of x from -oo to +oo.
The value of g(x) is restricted however to -oo..-5 and +10..-oo. Combining -oo..+10.
There is no shortcut to finding the range of discontinuous functions. you have to evaluate the functions in the area of the discontinuities. Big Rat's graphs will help.
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