Determine Compositions of Functions

[Alaba27]

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!

 

[MarkFL]

Let's begin with the first one...what is \(\displaystyle g(-\pi)\)?

 

[Alaba27]

Let's begin with the first one...what is \(\displaystyle g(-\pi)\)?

I thought it was (0,1), but I'm not sure.

 

[MarkFL]

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]

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]

Yes! (Cool)

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

 

[Alaba27]

Yes! (Cool)

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

1! So it is (0,1)?

 

[MarkFL]

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]

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]

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]

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]

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]

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]

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.

 

[Fermat1]

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.

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

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

\(\displaystyle g(f(h(x)))=?\)
https://www.physicsforums.com/attachments/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 $g(f(h(x)))=2sin(2.9^x-3^x+1)$