find the component f and d , so that fog =H

H(x)=log-3 e^x

What is the significance of the -3 after "log"? You cannot have a log of a negative number.

By the way, what is "recalus"?

find the component f and g so that fog =H

H(x)=log3 e^x

To find the components f and g such that fog = H, where H(x) = log₋₃eˣ, we need to find two functions f(x) and g(x) such that (fog)(x) = H(x).

Here's how to find f and g:

Step 1: Start with the given function H(x) = log₋₃eˣ.

Step 2: Since the composition fog involves applying g first and then applying f to the result, we need to find g(x) first.

Step 3: The composition fog is given by (fog)(x) = f(g(x)). In other words, f(g(x)) = log₋₃eˣ.

Step 4: To find g(x), let's set g(x) = eˣ. This means that we will apply the exponential function eˣ first.

Step 5: Now, we have f(eˣ) = log₋₃eˣ, which implies that f(x) = log₋₃x. So, the function f(x) = log₋₃x.

Step 6: Therefore, g(x) = eˣ and f(x) = log₋₃x.

So, f(x) = log₋₃x and g(x) = eˣ are the components that satisfy fog = H, where H(x) = log₋₃eˣ.