Find f(x) and g(x) such that f(g(x))= square root of x over 6 square root of x+5 .
To find functions f(x) and g(x) such that f(g(x)) is equal to the given expression, we need to decompose the expression into nested functions.
Let's start by breaking down the given expression:
f(g(x)) = √(x) / (6√(x) + 5)
To simplify this, we'll assign a new variable to the denominator (6√(x) + 5) and then express it as a separate function:
Let h(x) = 6√(x) + 5.
Now we have:
f(g(x)) = √(x) / h(x)
From this, we can see that the function g(x) is simply x. So:
g(x) = x
Now, let's solve for f(x):
f(g(x)) = √(x) / h(x)
Substituting g(x) into the expression, we have:
f(x) = √(x) / h(x)
Therefore, we have found the functions f(x) and g(x):
f(x) = √(x) / [6√(x) + 5]
g(x) = x