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 decompose H(x) into f(g(x)).
Let's start by breaking down H(x):
H(x) = (6x + 9)^4
We can see that the outer function, f, will involve raising to the power of 4. So, let's assume that f(x) = x^4. Therefore, we have:
f(g(x)) = (g(x))^4
Now, we need to determine g(x). If we look inside the parentheses of H(x), we can see that it is (6x + 9). This suggests that g(x) is the function which produces (6x + 9):
g(x) = 6x + 9
Now, substitute g(x) into the equation for f(g(x)):
f(g(x)) = (g(x))^4
f(g(x)) = (6x + 9)^4
So, the functions f and g are defined as follows:
f(x) = x^4
g(x) = 6x + 9
To verify this, you can substitute any value of x into g(x), then take that result and substitute it into f(x), and it should be equal to H(x).
Note: It's important to keep in mind that there can be multiple answers for f and g that satisfy the equation f(g(x)) = H(x). In this case, we chose f(x) = x^4 and g(x) = 6x + 9 as a possible solution.