Range of a function

Range of a function is nothing but the set of values taken by the function as the input variable takes on different values over the Domain. Consider the function in the Figure 1 below. I have drawn the graph using the following website: "http://rechneronline.de/function-graphs/".

In order to find the range of values taken by the function or equivalently the output variable y, we may start by drawing the graph of the function over its entire domain. If we look at the graph below, then we find that the function has two vertical and one horizontal asymptote. The two vertical asymptotes are x = 0 and x = 1 respectively. Note that vertical asymptote always pass through values of x for which the function is not defined. 

A vertical asymptote is always a line parallel to y-axis passing through the value of x where the function isn't defined, but with a condition that as x approaches those values (0 and 1 in this case) from either side of them, the function takes on values larger and larger in magnitude thus approaching +∞ or –∞. It "may" approach +∞ from one side of that value of x or –∞ from the other side. In our case, as x approaches 1 (or 0) by taking up values larger than 1 (or 0), function approaches +∞ and as x approaches 1 (or 0) by taking up values smaller than 1 (or 0), function approaches –∞. Note that it doesn't mean that at whichever value of x the function is not defined, we will have a vertical asymptote passing through it. But for a simple (non-composite) function, a vertical asymptote will be passing through only that value of x where the function isn't defined.

A horizontal asymptote is always a line parallel to x-axis (say y = a) such that as x approaches either +∞ or –∞ , the value of the function f(x) → a (but never equals a). In our case below, the line y = 0 (i.e., x-axis) is the horizontal asymptote as the value of the function approaches 0 as x approaches either +∞ or –∞.

Asymptotes need not just be vertical or horizontal. They could be slanted. A line 'y = mx + c' is a slanted asymptote to a function (or curve of the function) if as x approaches either +∞ or –∞, the curve of the function approaches increasingly closer to the line 'y = mx + c' but never touches it. For an example of a slanted asymptote, look at the graph of the function in Figure 2 below. Here the line y = 2x + 3 is a slanted asymptote to the blue curve of the function f(x). Note that the function f(x) also has a vertical asymptote through x = –1, but it has no vertical asymptote through x = 0, even though the function is not defined at x = 0. We would study more about such cases when we do limits.

 Figure 1

Figure 2

Now suppose we are asked to find the range of a simple function and we have no access to any website to draw the graph of the function. By the way, the range of the function in Figure 1 and Figure 2 is (–∞ , +∞). How do we find the range of any function without any external help?

1. First of all find the domain of the function (which amounts to finding the values of x where the function is not defined). Let the domain D be the union of disjoint sets A, B, C, & so on such that if any two sets share a bound, then none could contain the common boundary point. For example, in Figure 1, the domain of the function is union of the intervals (–∞ , 0), (0 , 1), and (1, +∞). The domain of the function in Figure 2 is the union of the intervals (–∞ , –1), (–1 , 0), and (0, +∞). All these intervals are open (not including their finite bounds if they have one) on both sides. But (–∞ , 0) and (–∞ , –1) are bounded on the right side only and have 0 and  –1 as their (least) upper bound respectively. (1, +∞) and (0, +∞) are bounded on the left side only and have 1 and 0 as their (greatest) lower bound respectively. (0 , 1) and (–1 , 0) are bounded from both sides. Note that the function in Figure 1 is not defined at x = 0 and 1, and the function in Figure 2 is not defined at x = –1 and 0, therefore no set contains the common boundary points.
2.  Now we have to check how the function behaves as x approaches +∞ and –∞ or any open bound of any set A, B, C in the domain (as defined above).
3. If as x → +∞ or –∞ or any open bound, f(x) → some fixed value (say L), then the function "may" never take the value L. To find out if there is some value of x, for which f(x) = L, we have to solve f(x) = L. If we are not able to find any such x value for which f(x) = L, then L is not in the range of the function. For example, in Figure 1, the line y = 0 is the horizontal asymptote (because limit of the function is 0 as x →+∞ or –∞), but if we solve f(x) = 0, we get x = 1/2 (function must be defined at this value). Thus y = 0 is in the range set of the function. Also for example in Figure 2, the function is not defined at x = 0 and limit of the function as  x → 0 is 2, but if we solve f(x) = 2, we get x = –3/2 (function must be defined at this value). Here note that we have used the fact that [(a/b) = k] ⟹ (a = bk) only if b ≠ 0. Thus y = 2 is in the range set of this function.
4. If however as as x → +∞ or –∞ or any open bound (from taking values on either side of it) and f(x) → +∞ or –∞, then the range would definitely be unbounded. 
5. If f(x) → +∞ as well as –∞, as x → +∞ or –∞ or any value where the function is undefined, then the range would be unbounded from left as well as right. If only one of the cases f(x) → +∞ or –∞ is true, then range would be bounded on one side. And if neither f(x) → +∞ nor –∞ as x → +∞ or –∞ or any value where the function is undefined, then range is bounded (from both sides). How to find these bounds of the value of the function? For that we need to find the global maxima and minima, if there are both upper and lower bounds of the range; maxima if only upper bound ; and minima in case of only lower bound.
6. To find out the global maxima or minima, you have have to check the values of the function at the boundary points of the domain, all stationary points (i.e., where f '(x) = 0), and all points where f '(x)is not defined.