Express in the form of f o g?

galwaystudent

I can do these functions when they are relatively straightforward but I am lost when they get more complicated.

For example:

Express the following fuctions in the form f o g:
a) F(x) = (cube root of x)/(1 + cube root of x)
b) G(x) = cube root of (x / (1 + x))

I dont even know where to start with these, especially the second one where it says G(x) instead of F(x). Any ideas?

LeixlipRed

First things first, the name is irrelevant. F(x), G(x), they're just names, they mean nothing. It could easily be ORANGE(x) or whatever, so don't let that confuse you.

What this question is asking you is to write the two functions given as a composition of two other functions. i.e. F(x)= f ○ g = f(g(x)).

I'll give you an example and see if you can apply it to your question. I have a function [latex]H(x)=4x^{2}[/latex]. You can see this as squaring something and then multiplying it by 4 (a composition of those two things). So let [latex]g(x)=x^{2}[/latex] and [latex]f(x)=4x[/latex] then f ○ g = f(g(x)) will be [latex]f(x^{2})=4(x^{2})=4x^{2}[/latex]. Or you could see H(x) as multiplying something by 2 and then squaring it and maybe you can work out what f and g would be in that case.

Fremen

Blegh... another question where someone had a specific idea in their head but phrased it badly.

You can come up with some fairly trivial answers for any old choice of F:

Let u(x) = x. Then F(x) = u o F(x).

and so on...

LeixlipRed

Yeh, exactly, this question is in every calculus textbook as well, pointless really.

galwaystudent

LeixlipRed wrote: »

First things first, the name is irrelevant. F(x), G(x), they're just names, they mean nothing. It could easily be ORANGE(x) or whatever, so don't let that confuse you.

What this question is asking you is to write the two functions given as a composition of two other functions. i.e. F(x)= f ○ g = f(g(x)).

I'll give you an example and see if you can apply it to your question. I have a function [latex]H(x)=4x^{2}[/latex]. You can see this as squaring something and then multiplying it by 4 (a composition of those two things). So let [latex]g(x)=x^{2}[/latex] and [latex]f(x)=4x[/latex] then f ○ g = f(g(x)) will be [latex]f(x^{2})=4(x^{2})=4x^{2}[/latex]. Or you could see H(x) as multiplying something by 2 and then squaring it and maybe you can work out what f and g would be in that case.

Yeah, I know how to do the straightforward ones like in your example. Its just in the ones I posted I can't see where you can 'pluck' the g(x) out of it. In your example, you let g(x) = X^2, however in the examples I posted there I cant see how I can pick something to assign to g(x)?

Fremen

Try looking at the "form" of the expressions for F and G, and look for places where you could substitute a Y for some term involving x. If the particular substitution you pick allows you to "get rid" of all the appearances of x, then you're home and dry.

For instance, if
F(x) = Sqrt( x^2 + a ^2), you could let Y = x^2, or x^2 + a^2.
Now your expression becomes

F(Y) = Sqrt(Y),

so the x^2 + a^2 function followed by the square root function gives you what you want.

galwaystudent

Fremen wrote: »

Try looking at the "form" of the expressions for F and G, and look for places where you could substitute a Y for some term involving x. If the particular substitution you pick allows you to "get rid" of all the appearances of x, then you're home and dry.

For instance, if
F(x) = Sqrt( x^2 + a ^2), you could let Y = x^2, or x^2 + a^2.
Now your expression becomes

F(Y) = Sqrt(Y),

so the x^2 + a^2 function followed by the square root function gives you what you want.

OK cheers. Here's what I have now for the first one -

a) [latex]F(x) = {\root 3 \of {x}/({1+\root 3 \of x } ) [/latex]

Let [latex]g(x) = \root 3 \of x[/latex]

Then [latex]f(g(x)) = g(x)/(1+g(x))[/latex]

Therefore [latex]f(x) = x/(1+x)[/latex]


Are they the correct values for f(x) and g(x)?
And the question reads asks me to 'Express the following fuctions in the form f o g:'

---

so what should my final answer look like syntactically?

Fremen

Yes, that's right. Any thoughts about the second one?

As for presenting an answer, you could just write "f(x) = blah, g(x) = blah" if it's clear from the context, or if you wanted to be completely unambiguous, you could write

F = f o g, where f = "..." and g = "..."

galwaystudent

Fremen wrote: »

Yes, that's right. Any thoughts about the second one?

As for presenting an answer, you could just write "f(x) = blah, g(x) = blah" if it's clear from the context, or if you wanted to be completely unambiguous, you could write

F = f o g, where f = "..." and g = "..."

Cheers. For the second one how does this look -

[latex]g(x) = x/(1+x)[/latex]
[latex]f(x) =\root 3 \of x[/latex]

Fremen

That's ok, yeah

jas2002

Let

Then

Therefore
...

Not sure if this is right - pls correct me if I'm wrong?. As LeixlipRed indicated and from memory, its simply a matter of substituting the value of g(x) into f(x) wherever an x appears.
See attachment - couldn't paste it into the text area!

Fremen

If F, f and g are as defined above, then F(x) = f(g(x)).

To see that what you're proposing isn't right, try plugging x=8 into F(x), and x=8 into your solution.
What you've written could be expressed as f(g(g(x)))

LeixlipRed

jas2002 wrote: »

Let

Then

Therefore
...

Not sure if this is right - pls correct me if I'm wrong?. As LeixlipRed indicated and from memory, its simply a matter of substituting the value of g(x) into f(x) wherever an x appears.
See attachment - couldn't paste it into the text area!

Commented on it in the other thread as well, we're not here to present solutions. Though in this case the person has already worked it out themselves.

Fremen

I think that was a quote with out the QUOTE box, with the comment below the ...