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)} =

How about

f(x) = 9-x
g(x) = 5^x
h(x) = x^2
f(g(h)) = f(g(x^2)) = f(5^x^2) = 9-5^x^2

Sure, here's one possible way to express the function H(x) = 9 - 5^(x^2) as f ∘ g ∘ h:

Let f(x) = x^2
Let g(x) = 5^x
Let h(x) = 9 - x

Then, f ∘ g ∘ h(x) = f(g(h(x))) = f(g(9 - x)) = f(5^(9 - x)) = (5^(9 - x))^2 = 5^(2(9 - x)) = 5^(18 - 2x)

So, H(x) can be expressed as f ∘ g ∘ h(x) = 5^(18 - 2x).

To express the function H(x) = 9 - 5^(x^2) in the form f ∘ g ∘ h, we need to find three functions f(x), g(x), and h(x) such that H(x) = (f ∘ g ∘ h)(x).

Let's start by treating the exponentiation part as a separate function, h(x) = x^2.

Next, we can define g(x) as the function that raised 5 to the power of its input, g(x) = 5^x.

Finally, we can define f(x) as the function that subtracts its input from 9, f(x) = 9 - x.

Now, we can express H(x) as the composition of these three functions:

H(x) = (f ∘ g ∘ h)(x)

H(x) = f(g(h(x)))

H(x) = (9 - (5^(x^2)))

To express the function H(x) = 9 - 5^x^2 in the form f ∘ g ∘ h, we need to find three functions f(x), g(x), and h(x) such that f(g(h(x))) = H(x).

Let's start by defining h(x). Since h(x) is the inner function, it should operate on the input first. In this case, we can define h(x) = x^2.

Now, we will define g(x). g(x) should operate on the output of h(x). Since the output of h(x) is x^2, we can define g(x) = 5^x.

Finally, we will define f(x). f(x) should operate on the output of g(x). Since the output of g(x) is 5^x, we can define f(x) = 9 - x.

Putting it all together, the expressions for f(x), g(x), and h(x) are:
f(x) = 9 - x
g(x) = 5^x
h(x) = x^2

So, the function H(x) can be expressed as f ∘ g ∘ h:
H(x) = f(g(h(x))) = (9 - (5^((x^2))))