I pretty much can find the domain of certain basic functions. I understand the domain and range in the form of the point (x,y) = (domain, range). However, I just don't comprehend the idea of finding the range of any function.
Here are two questions:
(1) Find the range of f(x) = -(sqrt{-2x + 3})
(2) Find the range of y = -x^4 + 4
The range is all such that the equation has a solution in the domain (i.e. ). This gives . In order for there to be a solution we need . Thus, . Which gives and this is of course in the domain. Thus, is the range.
The range here is any real number. Thus, we are asking to find for what is the equation solvable? This implies and in order to have a solution we need .(2) Find the range of y = -x^4 + 4[
Dr. Math put it this way:
"Domain and range are just two different words for "how far something extends"; specifically, a king's domain is the territory he controls, and an animal's range is the region it wanders through. So it makes some sense that the set of numbers a function "controls" would be called its domain, and the set through which its value can wander is called its range."
Merriam-Webster puts it this way:
Domain: a territory over which dominion is exercised ; the set of lements to which a mathematical or logical variable is limited; specifically : the set on which a function is defined. (The word comes from the Latin word for "lordship".)
Range: a place that may be ranged over; an open region over which animals (as livestock) may roam and feed; the region throughout which a kind of organism or ecological community naturally lives or occurs; the set of values a function may take on; the class of admissible values of a variable.
In mathematics:
The domain of a function f(x) is usually fairly easy to find. It is the set of all the numbers x that can be put into f(x) and have the result make sense. That is, the x values that don't make some expression inside a square root sign negative, or that don't make a denominator zero, and so on.
The range is the set of all values f(x) can take, as x takes every value in the domain.
Example:
As for the domain, there are no restrictions. You can assign any real number to x. So the domain is "all real numbers"
The range is the set of all values y can take, as x takes every value in the domain. In this problem, you know that the square of a number is greater than or equal to 0. Could y take the value -3?
If we try to solve
we get
which is impossible to solve, so -3 is not in the range. One way to find the values of y that are possible is to try solving the equation for x:
Now you can use the logic you used for the domain: what values of y will let this formula make sense? The radicand must be greater than or equal to 0.
Range =