TrigonometryComposition of functions!

Alaba27

New member
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.

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

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

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

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MarkFL

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

Alaba27

New member
Let's begin with the first one...what is $$\displaystyle g(-\pi)$$?
I thought it was (0,1), but I'm not sure.

MarkFL

Staff member
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
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

Staff member
Yes!

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

MarkFL

Staff member
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
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

Staff member
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
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

Staff member
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
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

Staff member
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
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

$g(-\pi)=0$ 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 $g(f(h(x)))=2sin(2.9^x-3^x+1)$