find the component f and g so that fog =H
H(x)=log3 e^x
To find the functions f and g such that fog = H(x) = log₃(eˣ), we need to break down the composition of functions.
Let's start by understanding the composition notation: fog(x) means applying the function g first, and then applying the function f to the result.
In this case, H(x) = log₃(eˣ). We can see that g(x) = eˣ since g(x) takes the exponent of x, and f(x) = log₃(x) since f(x) takes the logarithm base 3 of x.
Now, let's substitute these functions into fog(x) to see if it equals H(x):
fog(x) = f(g(x))
= log₃(eˣ)
As you can see, fog(x) is indeed equal to H(x) = log₃(eˣ).
Therefore, the functions f and g that satisfy fog = H are:
g(x) = eˣ and f(x) = log₃(x).