Express the function in the form f ∘ g ∘ h. (Use non-identity functions for f, g, and h.)
H(x) = 9 - (5)^x^2
{f(x), g(x), h(x)} =
To express the function H(x) in the form f ∘ g ∘ h, we need to find non-identity functions f(x), g(x), and h(x) such that H(x) = f(g(h(x))).
Let's break down the steps:
Step 1: Find the innermost function, h(x):
The function H(x) = 9 - (5)^x^2 already represents the innermost function. Therefore, h(x) = x^2.
Step 2: Find the intermediate function, g(x):
To find g(x), we need to determine what happens to x^2 in the function H(x). Looking at H(x) = 9 - (5)^x^2, we see that the exponent x^2 is being evaluated by the exponential function with the base 5. Therefore, let's set g(x) = 5^x.
Step 3: Find the outermost function, f(x):
To find f(x), we need to determine what happens to 5^x in the function H(x). Looking at H(x) = 9 - (5)^x^2, we see that the entire expression (5)^x^2 is being subtracted from 9. Therefore, let's set f(x) = 9 - x.
Putting all the steps together, we have f(x) = 9 - x, g(x) = 5^x, and h(x) = x^2. Thus, the function H(x) can be expressed as H(x) = f(g(h(x))), which is:
H(x) = (9 - (5)^((x^2)))
this nice article might help. If not, come on back and indicate where you get stuck...
https://www.purplemath.com/modules/fcncomp4.htm