Composition of a Function

[b1ackplague]

Find the formula for f composed of g, and g composed of f for f(x)=x/1+x^2 and g(x)=1/x?

I also need to state the domain which i think is just x cant = 0. I came to the solution x/x^2+1 for (FoG). I have no idea how to do (GoF). Please Help.

 

[mmm4444bot]

fog(x) is f(1/x).

gof(x) is g(x/[1 + x^2])

In other words, to find the definition of fog(x), replace every x in f(x) with 1/x, and simplify.

Your result for fog(x) looks good.

To find the definition of gof(x), replace every x in g(x) with x/(1 + x^2), and simplify.

Please show your work, if you would like more help.

Cheers ~ Mark

MY EDIT: Deleted my previous result for fog(x) because I made a mistake.

 

[b1ackplague]

Here's a scan of the work i've done so far.

 

[mmm4444bot]

Hi Black Plague:

My earlier post was wrong because I made a mental mistake while determining fog(x). Please excuse me. (I should stop trying to do algebra in my head, but sometimes I'm too lazy to get a piece of paper and the walls are already full.)

Your work looks good for f?g(x).

f?g(x) = x/(x^2 + 1)

Now, let's look at the domain of f?g(x).

We start by considering the domain of the inner function, which is 1/x. I see that you noted "x not equal to zero". That's correct, for 1/x. We must keep this restriction.

Looking at x/(x^2 + 1), we see that the denominator is never zero for any x because x^2 can never equal -1. Therefore, there are no restrictions to the domain from this outer function.

The domain of f?g(x) is all Real numbers except zero.

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Your setup for g?f(x) looks correct, too.

Dividing 1 by a ratio always gives the reciprocal of the ratio. In other words:

1/(a/b) = b/a

You can use this fact to simplify your start for g?f(x).

To consider the domain of g?f(x), start by looking at restrictions from the inner function. If you find any, they must be added to restrictions from the outer function.

There are many sites providing examples on how to find the domain of composite functions. Google keywords: [find domain of composite function].

Here are two results.

http://www.sinclair.edu/centers/mathlab ... omains.pdf

http://mathforum.org/library/drmath/view/54599.html

Cheers ~ Mark

 

[BigGlenntheHeavy]

\(\displaystyle f(x) \ = \ \frac{x}{1+x^2}, \ domain \ = \ all \ reals, \ g(x) \ = \ \frac{1}{x}, \domain \ = \ all \ reals \ except \ x \ \ne \ 0\)

\(\displaystyle [gof](x) \ = \ g[f(x)] \ = \ g\bigg(\frac{x}{1+x^2}\bigg) \ = \ \frac{\frac{1}{x}}{1+\frac{1}{x^2}} \ = \ \frac{x}{x^2+1}\)

\(\displaystyle Hence, \ [gof](x) \ = \ \frac{x}{x^2+1}, \ what \ is \ its \ domain?\)

 

[b1ackplague]

Thank you for all the help

The only domain restriction for [GoF](x) is x can't equal 0. Is that correct? Because you can plug any number into x^2 and get a positive or 0.

 

[mmm4444bot]

The only domain restriction for [GoF](x) is x can't equal 0. Is that correct? No.

g?f(0) exists.

Did you try to evaluate it ?

What is the domain of the inner function f(x) ?

What is the domain of the outer function x/(x^2 +1) ?