suppose H(x) = (1-7x)^4
find two functions F and G such that (f•g)(x)=H(x)
neither function can be identity function
g(x) = 1-7x
f(x) = x^4
h(x) = f(g(x))
To find two functions F and G such that (F•G)(x) = H(x), where H(x) = (1-7x)^4, we can express H(x) as the composition of two functions.
Let's assume F(x) = (1-7x)^a and G(x) = (1-7x)^b, where a and b are positive integers.
Using the properties of function composition, we have:
(F•G)(x) = F(G(x)) = (1-7G(x))^a
Substituting the expression for G(x), we have:
(F•G)(x) = (1-7[(1-7x)^b])^a
To obtain H(x) = (1-7x)^4, we need to determine values of a and b that satisfy this equation.
Comparing the exponents, we have:
a*b = 4
Let's find suitable values for a and b:
One possible solution is a = 1 and b = 4, as 1*4 = 4.
Therefore, we can express H(x) = (1-7x)^4 as the composition of two functions:
F(x) = (1-7x)^1
G(x) = (1-7x)^4
Using these functions, we have:
(F•G)(x) = F(G(x)) = (1-7[(1-7x)^4])^1 = (1-7(1-7x)^4)^1 = (1-7+49x)^1 = (1+49x-49)^1
So, (F•G)(x) = (1+49x-49).
Note: Both F(x) and G(x) are not the identity function.
To find two functions F and G such that (F•G)(x) = H(x), we need to express H(x) as the composition of two functions. Let's break it down step by step.
1. Start with the given function: H(x) = (1 - 7x)^4.
2. Let's rename (1 - 7x) as a new variable, let's say u. So, u = (1 - 7x).
3. Now, we can rewrite H(x) as H(u) = u^4.
4. From here, consider taking a square root. This will lead us to the composition of two functions: H(u) = (g◦f)(u), where f(u) = u^2 and g(u) = u^2.
5. Finally, substitute the original expression of u back into f and g to find the functions F(x) and G(x):
F(x) = (1 - 7x)^2
G(x) = (1 - 7x)^2
So, (F•G)(x) = F(G(x)) = H(x) = (1 - 7x)^4.
It's important to note that neither F(x) nor G(x) can be the identity function, which is the function that returns the same value it is given as an input. In this case, both F and G involve non-identity operations (squaring), making sure they are not identity functions.