Determine Compositions of Functions

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    Composition Functions
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Discussion Overview

The discussion revolves around determining compositions of functions as part of a homework problem. The functions involved are polynomial, trigonometric, and exponential, specifically focusing on evaluating expressions like $$f(g(-\pi))$$, finding the inverse of a function, and composing functions together without simplifying the results.

Discussion Character

  • Homework-related

Main Points Raised

  • One participant expresses difficulty with the homework problem involving the functions $$f(x)=2x^2-x+1$$, $$g(x)=2\sin(x)$$, and $$h(x)=3^x$$.
  • Another participant suggests starting with calculating $$g(-\pi)$$ and questions its value.
  • There is a discussion about the value of $$g(-\pi)$$, with some confusion over notation and the correct interpretation of the output.
  • Participants clarify that $$g(-\pi)$$ evaluates to 0, leading to the evaluation of $$f(0)$$.
  • There is a request for assistance in finding the inverse of the function $$h(x)$$, with a later response indicating the need to convert from exponential to logarithmic form.
  • One participant mistakenly suggests the inverse of $$h(x)$$ is the cube root, which is corrected by another participant.
  • Participants express a desire to understand the process rather than just receive answers, emphasizing the importance of grasping the concepts involved in function composition.

Areas of Agreement / Disagreement

Participants generally agree on the steps to evaluate the functions, but there is some confusion regarding the notation and the correct approach to finding the inverse of the exponential function. The discussion remains unresolved regarding the complete evaluation of all expressions.

Contextual Notes

There are limitations in the discussion regarding the assumptions made about function notation and the process of finding inverses, as well as the need for clarity in mathematical expressions.

Alaba27
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This question is killing me. I'm finding it difficult to do and it's a problem with my homework.

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

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

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

$$g(f(h(x)))=?$$
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View attachment 847

I am so lost right now. Please help!
 

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Let's begin with the first one...what is $$g(-\pi)$$?
 
MarkFL said:
Let's begin with the first one...what is $$g(-\pi)$$?

I thought it was (0,1), but I'm not sure.
 
You are only interested in the value the function returns not a point in the plane.

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

What is $$\sin(\pi)$$ ?
 
MarkFL said:
You are only interested in the value the function returns not a point in the plane.

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

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

That's 0.
 
Yes! (Cool)

So now, what is $$f(0)$$ ?
 
MarkFL said:
Yes! (Cool)

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

1! So it is (0,1)?
 
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:

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

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

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

Now for the second. We need to find $$h^{-1}(x)$$. 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. :(
 
  • #10
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?
 
  • #11
MarkFL said:
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.
 
  • #12
We need not worry about the composition yet, all we need first is to find the definition of $$h^{-1}(x)$$. Can you find this?

Once we have it, then we will proceed to find the given composition.
 
  • #13
MarkFL said:
We need not worry about the composition yet, all we need first is to find the definition of $$h^{-1}(x)$$. Can you find this?

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

It's cube-root x.
 
  • #14
That would be correct if $$h(x)=x^3$$, but we have $$h(x)=3^x$$. You are going to need to convert from exponential to logarithmic form.
 
  • #15
Alaba27 said:
This question is killing me. I'm finding it difficult to do and it's a problem with my homework.

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

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

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

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

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