technically the domain is part of the definition of a function and you can have different domains with the same functional equation for example we can have: f: R- to R with f(x)=1/x and g: R+ to R with g(x)=1/x so if I tell you h(x)=1/x you have no information about the domain however in applied math people usually just use a functional equation to represent a function and implicitely set the domain as the biggest possible subset within the real numbers so in the case of our example the function h can accept any real number except 0, hence its biggest possible domain is R \ {0} such that: h: R \ {0} to R with h(x)=1/x so the general strategy is to look for problematic values that cause division by zero or similar issues (like square root from negative numbers) and to exclude them consider for example: f(x)=sqrt(1-x2 ) in this case the problematic values are all x such that: 1-x2 <0 iff 1 < x2 iff 1 < |x| so the domain of f is all real numbers except those with an absolute values bigger than 1, which means only values between -1 and 1 are allowed: f: [-1, 1] to R, f(x)=sqrt(1-x2 )

A general method would be this:
Let
For t to be real,

But , since is equal to the square root of a real number.
So the range of the function will be .

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A more specific method for :
First of all, range() .
The range of is if and if .
So, the range of will be the square root of bounds of intersection of and the range for .

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Here are some of the "common rules" for to be real:
1. If , .
2. If .
3. If
4. If (Didn't want to add this rule since it is very specific)

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To find domain of a function, , find for what values of , will be undefined/not real. To find range, the general method is to find in terms of and then find values of for which is not defined.

The domain and range of a function can be determined without a graph by analyzing the function's formula for any mathematical restrictions on x-values (domain) and calculating possible y-values (range) through various mathematical techniques. Finding the domain and range of a function without a graph involves understanding the mathematical rules and restrictions on the function's formula. For the domain, you have to consider which x-values can be legally plugged into the function without causing undefined situations like division by zero, taking the square root of a negative number, or any other operation that isn't allowed. As for the range, it is the set of all possible outputs resulting from substituting the domain into the function. One method is to analytically solve for y (output) in terms of x (input), and then determine the possible values of y based on the values of x in the domain. Another approach is to use calculus techniques like finding derivatives to determine the behavior of the function, which can give insight into its range. It is imperative to take particular features of a function into account, such as asymptotes, discontinuities, and intervals of increase or decrease, as they can greatly affect the domain and range. As you gain more expertise, you can often anticipate what reasonable domains and ranges might be before even starting the calculation, much like setting expectations.

Set $$y=\frac{3x^2}{x^2-1}$$ and rearrange to get $$(y-3)x^2-y=0$$ This quadratic has real roots provided $b^2-4ac\geq 0$ which translates as $$y(y-3)\geq 0$$ Noting that $y=3$ is the horizontal asymptote, the solution set, i.e.the range, is $$y\leq 0, y>3$$

The best way to find the domain of a function is to find the values of x that are not in the domain and throw those values away. Once you've done that, whatever values of x are left give the domain. So you want to identify bad values of x and throw them away. What is a bad value of x? It's a value of x that causes some kind of problem. Common problems include: zero in the denominator of a fraction; a negative number under a square root; a nonpositive number inside a logarithm. Once you've identified all of the bad values of x and thrown them away, then the domain is everything that's left. === But what would I do in the case that have I something that isn't a fraction, like this: x + 2 = 1 + sqrt(2 + x) This doesn't make sense. That's an equation. It only makes sense to talk about domains of functions, not equations. or x - sqrt(2 - x) Three steps: Identify all of the bad values of x. Throw away those bad values. Everything that's left is the domain. So, what are the bad values of x here?

Between the points (-2,0) and (2,1) the base of the power is in (0,1).

The exponents have $x^2<1$ , while $2|x|+3 >1$.

It follows that the first term is greater than the second, because $$ \forall \ (x,p,q),\quad 0<(x,p)<1, q>1 \quad x^p>x,\quad x^q<x$$.