find the functions f and g so that f of g =H. H(x)=(6x+9)^4
please show work so i understand this better. thank you
To find the functions f and g such that f(g(x)) = H(x) = (6x + 9)^4, we need to separate the composition of functions into individual steps.
Let's start by defining the function g(x). Since g(x) is an intermediate function, we can choose any function that transforms x into something else. In this case, let's choose g(x) = 6x + 9.
Now, we need to find the function f(x) that takes the output of g(x), which is (6x + 9), and raises it to the power of 4. So, f(x) = x^4.
Now, to verify if f(g(x)) is equal to H(x) = (6x + 9)^4, we substitute g(x) into the function f(x):
f(g(x)) = f(6x + 9) = (6x + 9)^4
Thus, f(g(x)) = H(x) = (6x + 9)^4.
In summary:
g(x) = 6x + 9
f(x) = x^4
So, the functions f and g that satisfy f(g(x)) = H(x) = (6x + 9)^4 are:
f(x) = x^4
g(x) = 6x + 9