Stuck on a composition

[jinx24]

I've gotten a pretty good start on this problem, but I'm not really sure where to go from here.

"Let f(x) = (1/x^2) +1 and g(x) = 1/(x-1). Compute (f o g)(x)"

f[g(x)] = 1/(1/(x-1))^2

= 1/(1/(x^2+1)

From here I am not sure what to do. I am thinking I multiply by the reciprocal to get:

x^2 + 1 Am I headed in the right direction?

The second part of this problem is asking for the domain of (f o g)(x). How do I go about that? Is it just from my answer, or do I need to go back and find the domain of f(x) and g(x)?

As always, thank you.

 

[Unco]

Be careful, Jenny.

(x - 1)^2 doesn't expand to x^2 + 1.

And you dropped the 1: f(g(x)) = 1/(1/(x-1)^2) + 1

Ok.

You're quite right -- we could think of it as multiplying by the reciprocal:

\(\displaystyle \L\mbox{ \frac{ 1 }{\left(\frac{ 1 }{ a }\right)} = 1 \times \frac{a}{1} = a}\)


f(g(x)) is its own function. Call it h(x) if you like. h(x)'s domain can be determined by considering its equation alone. I bet you can recognise what the graph will look like. So what's its domain?

 

[jinx24]

Thank you Unco for catching me on my mistake. (x - 1)^2 does not expand to x^2 + 1. Instead, it expands to (x^2 - 2x + 1).

Ok. I am really having trouble with this problem. The instructions say to find (f o g), then state the domain, and then graph the function (f o g). I think I am close though.

= 1 / (1/(x-1))^2 + 1

= (x - 1)^2 + 1 (I multiplied by the reciprocal)

At this point, I don't know whether to keep it in this form or expand it to
(x^2 - 2x + 2). I think for the purpose of graphing, I should leave it at is, so that I can clearly see how to shift the graph.

2) Domain: Unco, you said that I could think of the composition as its own function, h(x). I trust everything you say, but I think you might be wrong here (but only by what I have found on my online textbook.) Forgive me if I am wrong.

According to my book, "The domain of (f o g) consists of those inputs x (in the domain of g) for which g(x) is in the domain of f."

I believe that I have to find the domain of g(x) first.
Domain f(x) = x cannot equal 1;
So the domain for (f o g)(x) is all real numbers except 1

I think I can handle the graph from here.

Sorry for such a long post!
Thanks a lot!
Jenny

 

[Unco]

Right on both counts.