Posted by Alissa on Sunday, December 18, 2011 at 12:45am.
g(x) takes input of "x" and gives output of "cuberoot(x - 5)" (that's what the little 3 means outside the root sign).
Imagine feeding that into the f function.
Note that f(anything) = (anything)^3 + 5.
So suppose we put in what we got from doing g(x). What we got was "cubert(x - 5)". So let's put that into the f function.
f(cubert(x-5)) = (cubert(x-5))^3+5
But (cubert(x-5))^3 is just (x-5).
And so (x-5)+5 is just x.
This is how you can show that f(g(x)) = x.
The nesting of parentheses helps you see the order in which the functions are applied. You read from the innermost out. So first function g gets applied to x and gives an expression that we can call g(x). Then the function f works on that g(x) as input and gives an output. Performing one function on the result of an earlier function is called composing functions, or composition of functions.
So you've seen how f(g(x)) = x.
Can you prove that g(f(x)) = x?
Start with x and apply function f to it. Then take the result and apply function g to that, using f(x) wherever "x" appears in the rule for g(x).
Not coming out right!!!!!!