# Help functions domain and range

Hello, I don’t understand domain and range at all and I always get the qs wrong and I’ve watched videos and done multiple exam qs plz help I’m really stressing my exams are in 6 months!!
Original post by Alevelhelp.1
Hello, I don’t understand domain and range at all and I always get the qs wrong and I’ve watched videos and done multiple exam qs plz help I’m really stressing my exams are in 6 months!!

The domain includes all the possible values of x. The range includes all the possible values of y.
So, for example, if we take the function f(x) = 5/x, we need to exclude zero from the domain as we are not allowed to divide by zero. When the function has a square root, for example, square root of (x+8), we need to have (x+8) positive. That means x>or equal to -8.
So in the domain, we only take values of x that are superior or equal to -8.
To easily find the values that can be in the range, find the inverse function and do the same as what I have explained for the domain.
I hope that helps
Basically what @Frogette says. But to get a bit technical here:

Domain could be anything, as long as every value in the domain can output something. So for instance, the domain of the function f(x)=x could very well be [1,2], because the function doesn't explode for every value of x between 1 and 2. If we talk in calculator terms, we don't get "ERROR" anywhere. Now in A-Level, it is understood that we want the largest possible domain. so clearly for f(x)=x, the largest possible domain is the reals.

Determining the domain is pretty straightforward. Start with the reals. If we see a denominator, take away the values of x that makes the denominator 0; if we see a square root, take away values of x that makes the inside of the square root negative, etc. Take away everything that's impossible, you'll left with the possible - i.e. the domain.

Range depends on the domain, but really it's just what can the function outputs. Recall domain could be anything. So for the weird case of f(x)=x with domain [1,2], the range would be [1,2]. But again, with A-Level, it is understood that the range would correspond to the largest possible domain. The tip would be to always find the domain first.

Now, finding the range could be tricky. The inverse method could be dangerous, because not every function has an inverse (e.g. f(x)=sin(x) for real valued x doesn't have an inverse), not all functions are normal looking, weird stuff happen when you take "inverses"*, etc.. I think this is the time to brush up your curve sketching skills. Make sure you know how every function should look like, how translation works, etc.. Then sketch a graph to "guess" what the range is. Range is often the hard part with a lot of intricacies - pretty much comes down to how many functions you know well. Desmos will be your friend for practice.

* For instance, what is the inverse of f(x)=x^2? We know it will be f^-1(x)=sqrt(x). But when you actually do the algebra, you'll quickly face a conundrum of whether we should take the plus or the minus square root. And to be careful here, f(x)=x^2 doesn't have an inverse over the reals in the first place.
Another one would be f(x)=x^3+sin(x)+2^x. The inverse would look really ugly here (possibly no formula for inverse either!). But you can "guess" both the domain and range should be the reals.
(edited 1 year ago)