Find functions f and g so that f o g = H.

H(x)=1/(x^2-5)

f(x) = 1/x

g(x) = x^2-5

f◦g = f(g) = 1/g ...

Sure, let's clown around with this! 🤡

Let f(x) = 1/x, and g(x) = x^2 - 5.

When we compose the functions f and g, we get:

(f o g)(x) = f(g(x)) = f(x^2 - 5) = 1/(x^2 - 5).

So, f o g = H, where H(x) = 1/(x^2 - 5), as desired! 🎉

To find functions f and g such that f o g = H, we need to decompose H into two functions and find their composition.

Let's start by breaking down H(x) into two functions, f and g:

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

We can rewrite H(x) as:

H(x) = 1/(x^2 - 5)

Now, let's substitute g(x) into the expression:

g(x) = x^2 - 5

Next, we'll find f such that it cancels out the denominator in H(x). We can do this by letting f(x) equal the reciprocal of x:

f(x) = 1/x

Now, we have:

H(x) = f(g(x)) = 1/(g(x))

Replacing g(x) with its definition:

H(x) = 1/(x^2 - 5)

Therefore, the functions f(x) = 1/x and g(x) = x^2 - 5 satisfy the equation f o g = H.

To find functions f and g that satisfy f o g = H, we need to decompose the function H(x) into its individual composite functions.

Starting with H(x), which is equal to 1/(x^2 - 5), let's break it down step by step:

1. First, let's determine the function g(x) such that g(x^2) = x^2 - 5.
- To do this, we isolate x^2 on one side of the equation: x^2 = g^(-1)(x^2 - 5).
- Since we want g(x), we take the square root of both sides: g(x) = sqrt(x^2 - 5).

2. Next, let's find the function f(x) such that f(1/x) = 1/x^2.
- Similarly, we need to isolate 1/x on one side of the equation: 1/x = f^(-1)(1/x^2).
- To get f(x), we take the reciprocal (invert) of both sides: f(x) = 1/(1/x^2).
- Simplifying the expression, we have f(x) = x^2.

3. Now, let's check if f o g = H by composing the functions:
- f(g(x)) = f(sqrt(x^2 - 5)).
- Substituting f(x) = x^2, we have (sqrt(x^2 - 5))^2.
- Simplifying, we get x^2 - 5.

Therefore, we can see that f(x) = x^2 and g(x) = sqrt(x^2 - 5) satisfy the equation f o g = H(x).