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Trigonometry Composition of functions!

Alaba27

New member
Apr 4, 2013
18
This question is killing me. I'm finding it difficult to do and it's a problem with my homework.

Given that \(\displaystyle f(x)=2x^2-x+1,\,g(x)=2\sin(x)\text{ and }h(x)=3^x\), determine the following. You need not simplify the expressions.

\(\displaystyle f(g(-\pi))=?\)

\(\displaystyle \left(h^{-1}\circ f \right)(x)=?\)

\(\displaystyle g(f(h(x)))=?\)



4512b3d77874cfcf3d6f6d2a8419dcc8.png
4c41e0c9ea3a2b45169fb1d8ed801f20.png

I am so lost right now. Please help!
 
Last edited by a moderator:

MarkFL

Administrator
Staff member
Feb 24, 2012
13,775
Let's begin with the first one...what is \(\displaystyle g(-\pi)\)?
 

Alaba27

New member
Apr 4, 2013
18

MarkFL

Administrator
Staff member
Feb 24, 2012
13,775
You are only interested in the value the function returns not a point in the plane.

\(\displaystyle g(-\pi)=2\sin(-\pi)=-2\sin(\pi)\)

What is \(\displaystyle \sin(\pi)\) ?
 

Alaba27

New member
Apr 4, 2013
18
You are only interested in the value the function returns not a point in the plane.

\(\displaystyle g(-\pi)=2\sin(-\pi)=-2\sin(\pi)\)

What is \(\displaystyle \sin(\pi)\) ?
That's 0.
 

MarkFL

Administrator
Staff member
Feb 24, 2012
13,775
Yes! (Cool)

So now, what is \(\displaystyle f(0)\) ?
 

Alaba27

New member
Apr 4, 2013
18

MarkFL

Administrator
Staff member
Feb 24, 2012
13,775
It is just 1, when you write (0,1) this notation means an ordered pair, usually representing a point in a plane. I would write:

\(\displaystyle f(g(-\pi))=f(0)=1\)

Now for the second. We need to find \(\displaystyle h^{-1}(x)\). Do you know how to find the inverse of a function and how to check your work to make sure you did it correctly?
 

Alaba27

New member
Apr 4, 2013
18
It is just 1, when you write (0,1) this notation means an ordered pair, usually representing a point in a plane. I would write:

\(\displaystyle f(g(-\pi))=f(0)=1\)

Now for the second. We need to find \(\displaystyle h^{-1}(x)\). Do you know how to find the inverse of a function and how to check your work to make sure you did it correctly?
Alright. But I don't really understand how to do the second question. The thing is that I have to be done this question within the next 20 minutes because my tutor is only going to be available for a little bit today. :(
 

MarkFL

Administrator
Staff member
Feb 24, 2012
13,775
Well, we best get busy then...and I can't simply give you the answers because you have an impending deadline. My best advice is to ask for help earlier. I will be happy to stand in for your tutor to help you get these done, but I want to make sure you understand how to do them for yourself. Our goal is to make sure people gain a better understanding.

Do you know how to find the inverse of a function?
 

Alaba27

New member
Apr 4, 2013
18
Well, we best get busy then...and I can't simply give you the answers because you have an impending deadline. My best advice is to ask for help earlier. I will be happy to stand in for your tutor to help you get these done, but I want to make sure you understand how to do them for yourself. Our goal is to make sure people gain a better understanding.

Do you know how to find the inverse of a function?
I know, and I'd rather understand how to do the work instead of just getting answers! I know how to find the inverses of functions, but I've never done it in a composition function.
 

MarkFL

Administrator
Staff member
Feb 24, 2012
13,775
We need not worry about the composition yet, all we need first is to find the definition of \(\displaystyle h^{-1}(x)\). Can you find this?

Once we have it, then we will proceed to find the given composition.
 

Alaba27

New member
Apr 4, 2013
18
We need not worry about the composition yet, all we need first is to find the definition of \(\displaystyle h^{-1}(x)\). Can you find this?

Once we have it, then we will proceed to find the given composition.
It's cube-root x.
 

MarkFL

Administrator
Staff member
Feb 24, 2012
13,775
That would be correct if \(\displaystyle h(x)=x^3\), but we have \(\displaystyle h(x)=3^x\). You are going to need to convert from exponential to logarithmic form.
 

Fermat

Active member
Nov 3, 2013
188
This question is killing me. I'm finding it difficult to do and it's a problem with my homework.

Given that \(\displaystyle f(x)=2x^2-x+1,\,g(x)=2\sin(x)\text{ and }h(x)=3^x\), determine the following. You need not simplify the expressions.

\(\displaystyle f(g(-\pi))=?\)

\(\displaystyle \left(h^{-1}\circ f \right)(x)=?\)

\(\displaystyle g(f(h(x)))=?\)



View attachment 846
View attachment 847

I am so lost right now. Please help!
Here are the answers:
$g(-\pi)=0$ so $f(g(-\pi))=1$. $h^{-1}(x)=log_{3}(x)$ so $h^{-1}(f(x))=log_{3}(2x^2-x+1)$. Finall, $f(h(x))=2.9^x-3^x+1$ so $g(f(h(x)))=2sin(2.9^x-3^x+1)$