Composition of Functions and their Domain

Question

I have two functions. N(x) = √(2x-4) and P(x)=8-2√(20-2x). The composition that I need is P(N(x)). However, when I just plug the functions into each other, I get a weird answer that I know should be simplified somehow, but don't know how: P(N(x)) = 8-2√(20-2(√2x-4)). Then, I'm asked to find the domain of this function, and that's where I'm really stumped. I can plug it into an online calculator and get [5.5, 10], but I don't even know if that's correct, or, if it is, how to get to it. I'd really like to learn how to do this process properly for my test.

Answer 1

Answer 2

N(x) = sqr(2x-4)

P(x) = 8-2sqr(20-2x)

N(P(x)) = 8-2sqr(20-2sqr(2x-4))

domain is any x value that does not make a denominator =0 (which is not relevant to this problem) or any x that makes the value inside the radical negative. that eliminates x<2. x must be= or > 2

but you also have to worry about the radical in P(x)

20-2x inside the radical seems to mean x<or=10. x cannot be >10

2</=x</=10, or [2,10] in interval notation, but put that on hold for a while

that's most of the apparent restriction on the domain, but there's more. the combination of the two functions in the composition add on another restriction or lessen the restriction

20-2sqr(2x-4) cannot be negative or less than 0 so

2sqr(2x-4) cannot be greater than 20

sqr(2x-4) cannot be greater than 10

2x-4 cannot be greater than 100

2x cannot be greater than 104

x cannot be greater than 104/2 =52

x cannot be greater than 52

while this at first seems redundant as we already seemed to find 2</=x</=10

test an x value between 10 and 52, such as 20 or 50 or even 10

10 works, so does 20, so does 52

the new domain for the composite function is

2</=x</=52 or [2,52] in interval notation

which is less restrictive than if you looked at the domains of each of the 2 functions individually

20-2sqr(2x-4) inside the radical means

2sqr(2x-4) must be less than 20

sqr(2x-4) must be less than 10

2x-4 must be less than 100

2x must be less than 104

x must be less than 52

suddenly the domain of the composite paradoxically switched from 2</=x</=10 to

2</=x</=52 or [2,52]

[5.5,10] is part of the domain, but only a small part of it, but you can see where that 10 may have come from, given the restrictions placed by the individual functions, before doing the composite

another method is graph the composite function, then look at the graph to see what values x can take on. Use a graphing calculator