Hey guys. Here are some math problem I'm having trouble with. Try and answer any one and I'll be very very thankful.
1. Describe in general terms how to find the domain of the composite function f(g(x)) then illustrate with these functions f(x)=1/x^2-1 g(x)=sqroot(x+1)
2. There's a 2 cylinder water tank with a 4 ft diameter thats 1 ft. tall. When there's a hole in the bottom it drains at 5ft/sec^2. Write a function for the rate water drains as a function of t
3. The length "C" of the hypotenuse of a right triangle is a function of the lengths "a" and "b" of the legs.
a) state a rule for the function [a,b]
b) find c(3,4) and c(12,5)
1. So the domain of of f(x) is basically all real numbers except x= 1,-1 and the domain of g(x) is (-1,infinity) but what i do not understand is what the domain of f(g(x) is.
2.So the formula for the volume of a cylinder is pir^2h. But how can this help??
and for 3....so c^2=a^2+b^2
the rule would be C(a,b)= a^2+b^2?
so would this mean that C(3,4) = 5? and C(12,5) = 13?
Your best approach with (1) in order to get understanding is to draw pictures.
This might help:
Definition:Composition of Mappings - ProofWiki
Work out what the domain of g is. Bear in mind f(g(x)) is "g first, then do f of what you get when you've done g."
Then work out what the image of g is (some call it "range", it's the set of numbers that can appear as the output of g).
Then the only numbers available for f to work on are the image of g.
But then there are numbers (-1 and 1) which can not be in the domain of f.
So these numbers must be removed from the image of g.
So any numbers in the domain of g which map to 1 and -1 must be removed for the domain of f and g for f(g(x)) to be a mapping.
So the domain of f(g(x)) is: all the numbers in the domain of g, removing the numbers that map to 1 and -1 (whatever they are, I haven't looked at what f and g actually are).
Does this help? It is understood that the concepts behind function theory are more complicated and fiddly than they look on the surface. Take time to study this stuff because it will save hours of confusion in the future.
The thing that bothers me about this problem is the information that "it drains at 5 ft/sec^2". The rate at which water drains from at tank should have units of "volume per time" or "ft^3/s". With a tank of uniform cross section, you could give the rate at which the height of water goes down, calling it (with "abuse of notation") the rate at which the water drains, but that would still be "ft/sec", not "ft/sec^2".