Addition or ordinates, product of functions, and composite functions

[EspoirTron]

Hey, so basically my book explains additions of ordinates, products of functions, and composite functions in a horrible way-therefore, I was wondering if someone could please explain to me how to go about adding ordinates both algebraically and by a visual approach (by sketching the two graphs) and thus sketching the resultant graph. Similarly, could somebody please explain how to figure out the product of two functions: algebraically and visually and thus sketching the graph of the product function. Lastly, could someone explain composite functions to me.
It is confusing me so much and I would greatly appreciate anyones help! Thanks in advance! 

[brightsky]

addition of ordinates:
y = f(x) + g(x)
sketch f(x) and g(x). run vertical lines down the page (lines should hit f(x) and g(x)). add the y-values to construct req graph. sometimes f(x) and g(x) have restricted domains, in which case the vertical lines you run down the page will not hit both f(x) and g(x). obviously you can't add one y-value to something that doesn't exist. this is why the domain of the new graph is given by the intersection of the domains of f(x) and g(x).

product of two functions:
y = f(x)*g(x)
same principle as addition of ordinates. draw f(x) and g(x), run vertical lines down the page, but this time MULTIPLY the y-values. a bit annoying, but still manageable. i've never seen an exam question that asks you to draw the product of two functions from 'scratch'.

composite functions:
y=f(g(x))=fog(x)
it is simply too hard to draw composite functions using the technique described above (i.e. by first drawing f(x) and g(x) and then trying to work out, for each x-value, what the corresponding y-value of the new functions will be). however you do need to know what composite functions are.
the best way to think of composite functions is by visualising g-machines and f-machines. so visualising a production line. in this case, the g-machine is in front of f-machine. now say you have a stack of x-values (literally imagine you have at hand all real numbers). now stuff these x-values one by one into the g-machine. but the g-machine is very selective and will not take certain x-values. say the g-machine doesn't like x-values less than 0. so every time you stuff an x-value less than 0 into the g-machine, it will reject it. the set of all x-values that the g-machine does accept is called the DOMAIN of g. now each time you input an acceptable x-value into the g-machine, it will spit out a new value according to some algorithm/rule. let us call this a g(x)-value. so you've tried every single x-value in the stack you have at hand and fed it into the g-machine. it rejected some, and accepted some. those which the g-machine accepted got converted into g(x)-values. the set of all g(x)-values is called the RANGE of g(x). now you take all the g(x)-values that the g-machine spat out and you feed it into the f-machine, which will convert them into f(g(x)) values according to another algorithm/rule. the set of all f(g(x)) values is termed the RANGE of f(g(x)). now here's the thing:the f-machine is, like the g-machine, really picky; and if the f-machine rejects ANY of the g(x)-values you feed into it, the whole process will stuff up, because no f(g(x)) value will outputted. so in order for f(g(x)) to be DEFINED, the range of g (the set of all g-values outputted by the g-machine) must be a SUBSET of the domain of f (ALL the numbers which the f-machine accepts). if this is not the case, then the function is UNDEFINED. if this IS the case, then we also know that the domain of f(g(x)) (all the x-values that are ultimately accepted and converted into f(g(x)) values) is EQUAL to the domain of g(x), since x-values which the g-machine accepted would have made it to the end, and been converted into an f(g(x)) value. whenever you are confronted with a composite function question, think this.

[EspoirTron]

addition of ordinates:
y = f(x) + g(x)
sketch f(x) and g(x). run vertical lines down the page (lines should hit f(x) and g(x)). add the y-values to construct req graph. sometimes f(x) and g(x) have restricted domains, in which case the vertical lines you run down the page will not hit both f(x) and g(x). obviously you can't add one y-value to something that doesn't exist. this is why the domain of the new graph is given by the intersection of the domains of f(x) and g(x).

product of two functions:
y = f(x)*g(x)
same principle as addition of ordinates. draw f(x) and g(x), run vertical lines down the page, but this time MULTIPLY the y-values. a bit annoying, but still manageable. i've never seen an exam question that asks you to draw the product of two functions from 'scratch'.

composite functions:
y=f(g(x))=fog(x)
it is simply too hard to draw composite functions using the technique described above (i.e. by first drawing f(x) and g(x) and then trying to work out, for each x-value, what the corresponding y-value of the new functions will be). however you do need to know what composite functions are.
the best way to think of composite functions is by visualising g-machines and f-machines. so visualising a production line. in this case, the g-machine is in front of f-machine. now say you have a stack of x-values (literally imagine you have at hand all real numbers). now stuff these x-values one by one into the g-machine. but the g-machine is very selective and will not take certain x-values. say the g-machine doesn't like x-values less than 0. so every time you stuff an x-value less than 0 into the g-machine, it will reject it. the set of all x-values that the g-machine does accept is called the DOMAIN of g. now each time you input an acceptable x-value into the g-machine, it will spit out a new value according to some algorithm/rule. let us call this a g(x)-value. so you've tried every single x-value in the stack you have at hand and fed it into the g-machine. it rejected some, and accepted some. those which the g-machine accepted got converted into g(x)-values. the set of all g(x)-values is called the RANGE of g(x). now you take all the g(x)-values that the g-machine spat out and you feed it into the f-machine, which will convert them into f(g(x)) values according to another algorithm/rule. the set of all f(g(x)) values is termed the RANGE of f(g(x)). now here's the thing:the f-machine is, like the g-machine, really picky; and if the f-machine rejects ANY of the g(x)-values you feed into it, the whole process will stuff up, because no f(g(x)) value will outputted. so in order for f(g(x)) to be DEFINED, the range of g (the set of all g-values outputted by the g-machine) must be a SUBSET of the domain of f (ALL the numbers which the f-machine accepts). if this is not the case, then the function is UNDEFINED. if this IS the case, then we also know that the domain of f(g(x)) (all the x-values that are ultimately accepted and converted into f(g(x)) values) is EQUAL to the domain of g(x), since x-values which the g-machine accepted would have made it to the end, and been converted into an f(g(x)) value. whenever you are confronted with a composite function question, think this.

Wow, thank you for that detailed response; I do appreciate it!
That makes a lot more sense now. I find that addition of ordinates and products of functions, is very tedious and can get a tad bit annoying at times. Although, I do appreciate you explaining it to me-as it all makes a lot more sense now! Thanks 

[Daenerys Targaryen]

The machines :O Is this something most schools teach?
I thought my teacher got high then made this up then taught it to us.
Its a very good way of thinking about it though.

[silverpixeli]

My teacher lent me the essentials textbook before the first SAC, which explains it with the machines. Maths Quest doesn't do a very good job.. Before reading the machines explanation I had something very similar in my mind, but the use of function machines is a very nice analogy and useful for explaining composite functions to my friends who ask. At our school, at least, people have a very limited knowledge of what a function is. They think it's some magic operation when really it's just a process that turns an input into an output based on some rule.. I guess I understand this better than most of the other students at my school because of a background in programming, but this sort of thing should actually be taught. It may be at other schools, but it wasn't taught to me.