Ok first let me just get straight what we mean by the output of one function becoming the input of another as pertaining to the present problem! So I'm thinking we have the range of f(x^2 - 1) being the "output", that is, outputting values between -1 and 0; and then these values become an "input" in the sense that they're used to determine the interval over which we need to then consider the illustrated function (essentially 1 - |x|), this interval being, accordingly, -1 to 0. Is that correct usage of the term input? It seems a little skewiff, since we're not technically putting those values back in, but merely referring to them for guidance during the subsequent stage of the operation. That's what I meant by "mapping", that the range -1 to 0 "reveals" the relevant domain of the illustrated function.
The way you emphasize the word "choosing" might just get me out of this rut I'm in, I get the feeling I've formed an inaccurate conception that I'm stubbornly clinging to; I'd been imagining the whole operation being sort of deterministic.
I suppose what I "object" to is the way that the sequence of stages of the whole operation seem kind of disjointed, awkward and counter-intuitive, but I guess I'm just not understanding the underlying ideas deeply enough.
I'm afraid I can't very well pinpoint the part I don't understand, I'm hoping you might be able to pick something out from what I've said here as at least a clue to the root of my misconception! If not, thanks anyway for trying!
This should be very helpful; at least it gives me more to talk about. Some of this will be about using words right, because that is a common stumbling block, but also because I have to make sure we understand one another. I'll mostly be just restating what you are saying, with some corrections or clarifications that may or may not scratch your itch.
First, the words "input" and "output", while not quite technical terms, are useful. For me, the input of a function is a specific value you put in, and the output is a specific value that results. So the
output of f(x^2 - 1) is not the
range (which is the
set of all output values), but the number that, in this problem, will turn out to be x^2.
I want to give a name to x^2 + 1 to make it easier to talk about it; let's say
g(x) = x^2 - 1. So the integrand is a
composite function, f(g(x)). The variable of integration is x; that is used as input to the function g, producing g(x) = x^2 - 1; then that is put into the function f, which in effect means we let t = x^2 - 1, and then put this number t into the function f shown on the graph.
Now we just follow the numbers -- "plumbing", as I said, because composition of functions is like piping the output of one machine to the input of the next. For the integral, x runs from -1 to 1; that's the input of g. For those inputs (the domain of g), we find that the output of g is between -1 and 0 (the range of g). In this problem
the output of g is fed into the input of f. (I emphasized the word "chosen" because this occurs merely because that's the way they wrote this problem, not something general.) So we now have numbers between -1 and 0 going into f. But we find that
for those inputs, f is defined by f(t) = t + 1. Here I'm using t rather than x, just as the author of the solution did, to avoid confusion between two uses of "x". (Note that the name of the input variable can be changed at will; it's just a "place-holder".)
We have this:
(x) --> g --> (t) --> f --> (integrand)
That is, the variable of integration, x, is processed by the function I'm calling g, producing an intermediate value I'm calling t, which then goes into the function f, whose output is what we are integrating.
So now, carrying out the composition by replacing t with g(x), we have
f(x^2 - 1) = (x^2 - 1) + 1
which simplifies to x^2. That's what the integrand is; the rest is just calculus. (Yes, the algebra and the juggling of multiple functions and variables are the hard part.)
Now, there was no need to mention it, but we can write a piecewise formula for all of f:
\(\displaystyle f(t)=\left\{\begin{matrix}
0;\quad -2\le t\le -1\\
t+1;\quad -1\lt t\le 0\\
1-t;\quad 0\lt t\le 1\\
0;\quad 1\lt t\le 2
\end{matrix}\right.\)
In general, you would have to find the intervals in x for which t = g(x) is in each interval of this definition, and replace f with its appropriate formula in each. What has happened in this problem is that the entire range of g (that is, all output values for the allowed inputs) lies within the interval [-1,0], so only the second formula is ever needed.
A few comments about what you said, which is probably mostly right.
You said, "we're not technically putting those values
back in". What we're doing is putting them into the
next function, not
back into the
same function. That may be part of your problem. We're just moving forward through the plumbing, step by step. I sometimes draw the plumbing diagram I showed above, to help me keep track of things.
You said, "the sequence of stages of the whole operation seem kind of
disjointed, awkward and counter-intuitive". That may be because you are expecting something routine, when as I've said, the problem is designed to make you think at each step, not just do things you've done over and over. It's intentionally different, to make sure you are not just following a rut, but are focusing your attention on the meaning of each thing you do. Again, this is in the design of the problem, not in the nature of the math. It's like an obstacle course where you have to go slowly and watch your feet to avoid tripping.
But if you can get through it (getting up off the ground a few times and starting over), you know better where your feet are, even when you're running straight.
Does that help at all?
