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# C3: finding functions ranges and domains watch

1. How do you find ranges for functions? I get the impression its just guess work, or is there a method behind the madness?

Same goes for domains...
2. (Original post by riddles55)
How do you find ranges for functions? I get the impression its just guess work, or is there a method behind the madness?

Same goes for domains...
Umm. It is definately not guess work although it does sometimes seem that way to me.

The domain of a function is the set of numbers that you start with before applying the function to them.

So in many cases this is any number. But not for example if the function is y = root(x) or if it is y = arcsin(x) where only specific values yield valid answers (i.e. can't be negative in first example and must be between -1 and 1 in second).

The numbers that are produced by the function are the range and often this is any number. But once again not always. For example cos(x).

If the domain of a function is restricted "artificially" (i.e. it is not necessary for it to be restricted in this way) so like y = 10x, x > 15. Then you have to carry the effect of the restriction onto the range. In this example that means that y > 150.

The only other domain / range thing that I havent mentioned is in fractions where the domain is often restricted to stop a division by zero.

e.g. y = 56/(1 - x), x ≠ 1.

Oh yes and one more thing: with logs (say y = lnx) x has always got to be bigger than zero. So the domain in restricted to x > 0 but the range is still any number (think of the y = lnx graph).

Conversely with y = e^x the domain is not restricted but due to the nature of the function the range is only positive values.

I think that is all you need to know. If ever in doubt about either the range or the domain it could well be any number at all.
3. The domain is just the possible values that can be put into the function
e.g. f:x |---> 1/(1-x), x E Z, -4≤x≤5, x≠1
and the range is the possible values that the function can generate, in this case -0.25<f(x)<0.2.
To find the range of linear functions just put the two values of the domain in.
For non-linear functions you'll need other methods, like completing the square for quadratics.
You never get asked to find the domains of functions cos they need to be given in the question, but you may be asked to find the domain of the inverse function, f-1, having been given the domain of the original function.
4. You don't guess.

For range you either:
1) Use a characteristic of the function:
eg(for all real x):
e^x>0.
x^2>0
-1<=cosx<=1
-1<=sinx<=1
or
2) Change the function into a different form:
eg
f(x)=x^2+2x+3=(x+1)²+2, so f(x)>=2.

Note that differentiation and setting f'(x)=0 can also sometimes be effective but stationary points are only local minima/maxima and so the values of f(a) where f'(a)=0 may not be at the end points of the range.

For domain you need normally consider x to be real but exclude the values that cause f(x) to be non real.
eg:
f(x)=1/x(x-a)(x-b), then x E R but x≠0,a,b.
f(x)=lnx, then x E R but x>0.
f(x)=tanx, then x E R but x≠pi/2, 3pi/2, 5pi/2....
5. thanks everyone. One last question tho, what is the connection between range and domains for functions and their inverse? i cant rememeber the exact details.

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