Question

Could you please help with this question for domain, range, and end behavior? I started it, but I think I'm doing it wrong.

Answer 1

It looks like you have the domain confused for the range! You can think of the domain as the set of all "inputs" for a function (all of the x values which are allowed). In the given function, we have no explicit restrictions on the domain, and no situations like division by 0 or taking the square root of a negative number that would otherwise put limits on it, so our domain would simply be the set of all real numbers, R. Inequality notation doesn't really use ∞, so you could just put an R to represent the set. In set notation, we'd write

$\{x\in\re\mathbb{R}\}$

and in interval notation,

$(-\infty,\infty)$

The range, on the other hand, is the set of all possible outputs of a function - here, it's the set of all values f(x) can be. In the case of quadratic equations (equations with an x² term), there will always be some minimum or maximum value limiting the range. Here, we see on the graph that the maximum value for f(x) is 3. The range of the function then includes all values less than or equal to 3. As in inequality, we can say that

$f(x)\leq3$,

in set notation:

$\{f(x)\in\mathbb{R}\ |\ f(x)\leq3\}$

(this just means "f(x) is a real number less than or equal to 3")

and in interval notation:

$(-\infty,3]$