1. ## Combinations of Functions

I am having a fair bit of trouble with combinations of functions.

First, I'd just like confirmation on whether or not I done a more basic question correctly.

1.) If $f(x)=x^2 -x -2, g(x)= x - 3$ ,and $h(x) =2x$ determine the following:

b.) $f(h(g(0)))$

So
$g(0)=0-3$
$g(0)=-3$
$h(g(0))=2(-3)$
$h(g(0))=-6$
$f(h(g(0)))=f(-6)$
$f(h(g(0)))=(-6)^2 -(-6) -2$
$f(h(g(0)))=40$

Is this correct?

For the next question I am given the following:

I don't know how I am supposed to use the graph to evaluate each expression here. Also, is the expression for a.) the same thing as $f(g(2))$?

If $f(x)=x^2 -x +2$ and $g(x)=x-2$, find $h(x)$ such that $f(x)=g(h(x))$.

I have no idea how to solve this one.

2. ## Re: Combinations of Functions

Originally Posted by BobRoss
For the next question I am given the following:
The first is correct.
BUT the second is based on the two graphs.
a) $g(2)=0$ so from the graph $f(0)=~?$.

3. ## Re: Combinations of Functions

Sorry I'm still confused. Should I be looking at the x or y values on the graph?

4. ## Re: Combinations of Functions

You have to use that: (f o g)(x)=f(g(x))

5. ## Re: Combinations of Functions

Okay, so is the answer for a.) $f(0)=4$?

6. ## Re: Combinations of Functions

Originally Posted by BobRoss
Sorry I'm still confused. Should I be looking at the x or y values on the graph?
When $x=2$ then $g(2)=0$ so from the graph $f(g(2))=f(0)=1$.

7. ## Re: Combinations of Functions

Okay I think I see now. So for b, $f(2)=3$ then $g(3)=1$

And for c, $f(1)=2$ then $f(2)=3$

Are these correct? Now what do I do for d?

8. ## Re: Combinations of Functions

Yes, those are correct. For (d) (f+ g)(2), use the definition, which I am sure you were given: f+ g is defined by (f+ g)(x)= f(x)+ g(x). From your graphs determine f(2) and g(2), then add those values.

9. ## Re: Combinations of Functions

d) Note that $(f+g)(2)=f(2)+g(2)=...$

@HallsofIvy:
Sorry, I didn't see your post.

10. ## Re: Combinations of Functions

Okay so then since $f(2)=3$ and $g(2)=0$ then $f(2)+g(2)=3+0=3$ ?

I have two more questions that I am stuck on. They are on the assignment but I can't find any examples of them in the text so I don't know how to approach them.

1.) if $f(x)=x^2 -x+2$ and $g(x)=x-2$, find $h(x)$ such that $f(x) = g(h(x))$

2.) If $g(x)=x-2$ and $f(x)=3x-2$, find x such that $g(g(x))=f(f(x))$

11. ## Re: Combinations of Functions

1) What is $g(h(x))$ if you know that $g(x)=x-2$? Can you find $h(x)$ now?

12. ## Re: Combinations of Functions

Originally Posted by BobRoss
Okay so then since $f(2)=3$ and $g(2)=0$ then $f(2)+g(2)=3+0=3$ ?
1.) if $f(x)=x^2 -x+2$ and $g(x)=x-2$, find $h(x)$ such that $f(x) = g(h(x))$

2.) If $g(x)=x-2$ and $f(x)=3x-2$, find x such that $g(g(x))=f(f(x))$
I will do 1) if you will 2) and post the solution.
$h(x)=x^2-x+4$

13. ## Re: Combinations of Functions

Gah, I still don't really know for either question. I just can't seem to wrap my head around combinations of functions, especially when working backwards through them. For the second question, what I've done so far is:

$g(x-2)=(x-2)-2$
$g(x-2)=x-4$

$f(3x-2)=3(3x-2)-2$
$f(3x-2)=9x-6-2$
$f(3x-2)=9x-8$

That doesn't seem to be correct though. What have I done wrong?

14. ## Re: Combinations of Functions

Originally Posted by BobRoss
Gah, I still don't really know for either question. I just can't seem to wrap my head around combinations of functions, especially when working backwards through them. For the second question, what I've done so far is:

$g(x-2)=(x-2)-2$
$g(x-2)=x-4$

$f(3x-2)=3(3x-2)-2$
$f(3x-2)=9x-6-2$
$f(3x-2)=9x-8$

That doesn't seem to be correct though. What have I done wrong?
Set them equal to one another and solve.

15. ## Re: Combinations of Functions

Oh of course, I don't know why I didn't think of that. So then:
$x-4 = 9x-8$
$x=1/2$

And inputting that value of x into the equations they are equal. So that is right? Now I'm still fairly confused for the first question and don't know where to begin with it.

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