Why graph a function?

Please enlighten me as to how graphing a function helps. I can see a graph's utility with simple functions as they instantly give you value of dependent variable. But ignoring them and considering three dimensional complex figures that often arise in higher level math, how does it help you solve the problem if you know that the graph is a knot or a doughnut or in 2D case say the 'U' shape of a parabola. I admire the wonderful shapes but I don't know what to make of them except having a visual treat. How do they help you solve a given problem?

Edit: Take the following graph as example. Some of you may recognize it but for others who don't what information does it convey to you in absence of any equation and how will it help you understand the physical phenomenon.

Please take me through your mental journey. I am desperate to see the world with same intuitiveness as any one of you do. Thanks again for your time

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@jabba I think OP wants examples of problems that graphs help solve. – Austin Mohr Nov 6 '11 at 6:42
"Helps" what? There are certainly things for which graphing the function will probably not help much; but there are things where at least a sketch of the function is useful. If you need to determine global extremes of a function over a non-closed or non-finite interval, knowing the general shape of the graph can help you zero-in on where you should be looking; knowing that the function changes from concave up to concave down tells you something about the rate of change (which can be important; e.g., in budgetary discussions, it is important to know if the rate of change has reached a max). – Arturo Magidin Nov 6 '11 at 6:49
For the task of finding the roots of, or when integrating a one-variable function, you'll very often benefit from graphing the function in question. The graphs will often transparently point to any computational difficulties you might subsequently encounter. – J. M. Nov 6 '11 at 9:46
Imagine a person which is so blind that she cannot even make and look at mental pictures of mathematical objects. What would $f(x):=\exp(-x^2/2)$ mean to such a person? – Christian Blatter Nov 6 '11 at 9:55
Just to add another thought, usually to draw the graph of the function we need to know a lot about it. Especially in the day of calculators, it might seem like drawing a graph is trivial, but calculators often mess up complicated graphs. The easy ones we already know a lot about so graphing them is easy. The point is, we can't really get information from the graph until we already know enough information about the function to graph it... at which point we don't need the info the graph provides. – Graphth Nov 6 '11 at 17:06

Here is a simple task that will be tough without graphing: In how many points is the function

$$f(x) = x-\frac{1}{2} \left\lfloor \frac{1}{2} \left(\sqrt{8 x-7}-1\right)\right\rfloor \left(\left\lfloor \frac{1}{2} \left(\sqrt{8 x-7}-1\right)\right\rfloor +1\right)$$

not continuous? From the formula its not obvious but if you look at the graph you will see that the function does something very simple:

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Thanks Listing for your response. I agree that it is not obvious from the equation but could be discovered through calculations. So do we draw a graph to save time? – R2D2 Nov 6 '11 at 16:19
Yes, but we also draw the graph to understand what the function actually "does". It is quite hard to see by calculations that it is the function that raises from $1$ to $k$ and then falls back to $1$ to raise to $k+1$ and so on. Of course everything you see with your eyes can also be seen by doing the calculations but it is hard to do the calculations when you don't know what you are looking for. – Listing Nov 6 '11 at 16:24
Thanks again. I am not sure I understand the implication of "does" here. Continuing with the same example, what have you gained by learning that the function rises from 1 to k and then falls back to 1 to raise to k+1? Does it have physical implications? Is there a example of a real physical event, the behavior of which is revealed by drawing the graph of its function. – R2D2 Nov 6 '11 at 16:30
You know that the limit for $x \rightarrow \infty$ does not exist, that the function is unbounded and has no global maximum, that its minimum is $1$ and so on. I cannot think of any physical interpretation right now. – Listing Nov 6 '11 at 16:38
It depends on what your definition of "is" is. – Graphth Nov 6 '11 at 16:58

I suppose it would depend on the problem, or even the type of problem. In some cases, you can't just look at the equation and see what the answer could be, yet after drawing it out, you can clearly see that the line climbs for a while, then levels off at some value. $That$ value may be what you're trying to find.

Other problems you may be looking for symmetry, which you $can$ find algebraically, but it is a lot easier to just draw it and $look$ at it to see if both sides match up.

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You can also get a sense for the number and general location of the roots, whether or not the function is monotonic, etc. The graph doesn't necessarily solve your problem, but it gives you good intuition for how to proceed. – user7530 Nov 6 '11 at 8:21
Thank you user7530. Unfortunately for me I find myself no different when I see a graph of logarithmic function or exponential function. I am trying to figure what others can see so clearly or intuitively but I am unable to discover. Any help is much appreciated. – R2D2 Nov 6 '11 at 16:33

For example if we are talking about $2$-D functions $y=f(x)$ you may use graph to test whether function is one-to-one as it is shown here.

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