Thread: help with functions

Find functions f and g such that h(x) = f(g(x)). Note: do not choose f(x) = x or g(x) = x

22. h(x) = ln(x^3) 23. h(x) = (ln x)^3


The answer to 23 is: f(x) =x^3
g(x) = ln x

but I dont understand how to get it. Sorry if this is incredibly simple, and thanks for any help

Originally Posted by Yamahammer342

Find functions f and g such that h(x) = f(g(x)). Note: do not choose f(x) = x or g(x) = x

22. h(x) = ln(x^3) 23. h(x) = (ln x)^3


The answer to 23 is: f(x) =x^3
g(x) = ln x

but I dont understand how to get it. Sorry if this is incredibly simple, and thanks for any help

Remember that the "x" in f(x)= some expression is a "place holder". f(x)= x^3, f(a)= a^3, f(b)= b^3. Whatever is in the parentheses on the left replaces the x in the formula. If g(x)= x^3, f(g(x))= f(x^3) and, with f(x)= ln(x), that becomes f(g(x))= f(x^3)= ln(x^3).

f(g(x)) means "first do g(x), then apply f to that". "first do g(x)" gives g(x)= ln(x) and, since f(x)= x^3, "apply f to that" means to cube it: (ln(x))^3.

Of course the answer to 22 is f(x)= ln x, g(x)= x^3 so that f(g(x))= f(x^3)= ln(x^3)