Final answer:
To express the function F(x) = (2x+x²)⁴ in the form f o g, define g(x) = 2x + x² and f(u) = u⁴, then f(g(x)) equals the original function F(x).
Step-by-step explanation:
To express the function F(x) = (2x+x²)⁴ in the form f o g, we first need to identify two functions, f and g, such that f(g(x)) is equivalent to the given function. We can consider g(x) to be the inner function that will be input to the outer function f. It is often helpful to look for a composite function as a function inside of another function.
Let's define the function g(x) as g(x) = 2x + x². We notice that F(x) is essentially the function g(x) raised to the fourth power. Therefore, we can define the function f(u) as f(u) = u⁴. Notice that u here is just a placeholder for whatever we input into the function f.
By substituting g(x) into f(u) we get f(g(x)) = (g(x))⁴, which is equivalent to (2x + x²)⁴. We can see from this that f o g is equal to our original function F(x).
To bring clarity to this process, remember that squaring of exponentials involves square the digit term as usual and multiply the exponent of the exponential term by 2. Also, any number raised to the fourth power is that number multiplied by itself four times.