Suppose h(x) = 3/x^2-1. Find two nontrivial functions such that
h(x)=(f o g)(x)
(f o g referring to f of g)
F(x)=√2x-1. Find non trivials of f and g such that (fxg)=F
I need answer
I need answer
To find two nontrivial functions f(x) and g(x) such that h(x) = (f o g)(x), we need to find functions f(x) and g(x) where g(x) is the input for f(x), and the output of g(x) is the input for h(x).
Let's break down the composition f o g in order to find the functions f(x) and g(x).
1. First, let's assume g(x) = x^2 - 1.
The function g(x) takes an input x, squares it, and then subtracts 1.
2. Now, let's find the function f(x) that takes the output of g(x) as its input.
We have h(x) = 3/(x^2 - 1).
To match this form, we need to find a function that takes the reciprocal of its input.
Let's set f(x) = 1/x.
Therefore, the two nontrivial functions that satisfy the equation h(x) = (f o g)(x) are:
g(x) = x^2 - 1
f(x) = 1/x
To verify this, we can substitute g(x) into f(x):
(f o g)(x) = f(g(x))
= f(x^2 - 1)
= 1/(x^2 - 1)
This matches the original function h(x) = 3/(x^2 - 1), so we have found the correct functions f(x) and g(x).