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Composite Functions A2 maths problem?

I'm doing some questions, and I am a bit confused as to how to work out the domain and range of composite functions.

I need the domain and range of fg(x)
however, I dont know how you work out the domain
I know the domain is the values you are able to input, which would be any number in this case, however, do you have to use the original domains for f(x) and g(x)??

(edited 4 years ago)

Reply 1

RDKGames

Study Forum Helper

Original post by mushed

I'm doing some questions, and I am a bit confused as to how to work out the domain and range of composite functions. This is my problem:
𝑓(𝑥) = cos 𝑥 , 0 ≤ 𝑥 ≤ 𝜋
𝑔(𝑥) = 𝑥 +𝜋/2, 𝑥 ≥ 0

I need the domain and range of fg(x)

I have so far worked out that fg(x)= cos(x+𝜋/2), however, how do you work out the domain?
I know the domain is the values you are able to input, which would be any number in this case, however, do you have to use the original domains for f(x) and g(x)??

The input into cos has to be between 0 and pi.

So set

0 < x+pi/2 < pi

Hence what are the bounds for x from this inequality??

Now also take into account that x>0 due to the second function.

Reply 2

mushed

Original post by RDKGames

The input into cos has to be between 0 and pi.

So set

0 < x+pi/2 < pi

Hence what are the bounds for x from this inequality??

Now also take into account that x>0 due to the second function.

Thank you! This may be a stupid question but how would you "take into account x>0" for the second function? Does this mean the bounds would be 0<x ? Sorry, I know this is probably super simple, but domains confuse me for some reason and at the moment I obviously don't really have a teacher to ask for help, and I don't want to fall behind!

(edited 4 years ago)

Reply 3

RDKGames

Study Forum Helper

Original post by mushed

Thank you! This may be a stupid question but how would you "take into account x>0" for the second function? Does this just mean the bounds would be 0<x<pi/2 ? Sorry, I know this is probably super simple, but domains confuse me for some reason and at the moment I obviously don't really have a teacher to ask for help, and I don't want to fall behind!

Yep, so you got the important part of disregarding all the values of x which are between -pi and 0 from solving the first inequality.

Reply 4

mushed

Original post by RDKGames

Yep, so you got the important part of disregarding all the values of x which are between -pi and 0 from solving the first inequality.

Thank you so much! This makes a lot more sense now :smile:

Reply 5

RDKGames

Study Forum Helper

Original post by mushed

Thank you so much! This makes a lot more sense now :smile:

Good. All we've done was first impose a condition on the input of outter function. This ensures that whatever is being inputted into the first function is a value between 0 and pi. Solving this condition yields the possible set of x values we can have.

But then you also need to realise thar the inner function has restrictions as well, so you need to take those into account. This yields another possible set of x values we can have.

In the end, your overall domain is the set of values which satisfies BOTH possible sets.