SOLUTION: Find f(x) and g(x) so that the function can be described as y = f(g(x)). y = 4/x^2 + 9

Click here to see ALL problems on Rational-functions

Question 1092039: Find f(x) and g(x) so that the function can be described as y = f(g(x)).
y = 4/x^2 + 9
Found 2 solutions by ikleyn, greenestamps:
Answer by ikleyn(52805)   (Show Source):
You can put this solution on YOUR website!
.
Is it   (4/x^2) + 9   or   4/(x^2 + 9) ??

-----------------------
Your formula is ambiguous and unreadable without using parentheses.

Use parentheses to make your formulas readable and unambiguous.

Use parentheses when you post your formulas to this forum.

It is the REQUIREMENT of this site.

There is NO OTHER WAY to make them readable and unambiguous.

By violating this RULE, you work against yourself and against your better interests.

With it, welcome to the forum !!

---------------
I repeat this mantra to each and every new visitor coming to this forum,
until he or she will take it seriously.

Answer by greenestamps(13200)   (Show Source):
You can put this solution on YOUR website!

There are many ways to do this. Two easy, but not very useful, ways are these:
and

or
and

In both of those examples, one of the functions does nothing to the input and the other does everything. But in both cases you get the right result:

To get a meaningful way to get the expression as the composition of two functions, think about how you would go about evaluating the expression for a given value of x. Probably, you would first divide 4 by the square of the input; and then you would add 9. So make those two operations your two functions:

and

That would give you

While it is far less likely, you might notice that 4/x^2 is (2/x)^2; so you could define the two functions as

and

That would give you

And there are probably other less obvious and more complicated ways to get the desired function as a composition of two functions.