Find two functions f and g such that (f ∘ g)(x) = h(x) (Using Non-identity functions f(x), g(x))

Question

How do you find h(x)= -x²+5/3-x^2

Answer 1

So if we want to find h(x), we're gonna need to understand what this means:

(f ∘ g)(x) = h(x)

I prefer to write it like this:

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

What this means is that h is a composition of functions. Essentially, to get h, we took some function (g), put in x to get g(x), and then plugged in g(x) into f to get f (g(x)). We can choose what we want for both f and g

To start off, let's simplify our equation, simply combine both -x² 's:

-x² + 5/3 - x² = -2x² + 5/3

To obtain h(x) there's literally so many ways we could do it. Here's one of the easier ways:

First we let g(x) = -2x²

Remember earlier in the bold I said, we took some function (which we can choose) and plugged g(x) into another function, f. Now that we have g(x), let's make up some f(x) that will make it combine into h(x).

If we choose f(x) = x + 5/3

How does this work? Well, h(x) is f(g(x)), and f(g(x)) is simply the function f, but you replace the x's in the equation with g(x). Let's see what that is:

h(x) = f(g(x)) = g(x) + 5/3 = -2x² + 5/3

So the question said to find (read: make up) two functions f and g so that f(g(x)) = -x² + 5/3 - x²

Welp, we found those two functions. They are g(x) = -x² and f(x) = x + 5/3