Final answer:
To find f ◦ g, substitute g(x) into f(x) and calculate the expression, resulting in (x + 2)² + 1, or x² + 4x + 5. To find g ◦ f, substitute f(x) into g(x) to get (x² + 1) + 2, which simplifies to x² + 3.
Step-by-step explanation:
To find f ◦ g, we substitute the function g(x) into the function f(x), which means we replace x in f(x) with g(x).
So, f ◦ g(x) = f(g(x)) = (g(x))² + 1 = (x + 2)² + 1 = x² + 4x + 4 + 1 = x² + 4x + 5.
To find g ◦ f, we substitute the function f(x) into the function g(x), which means we replace x in g(x) with f(x).
So, g ◦ f(x) = g(f(x)) = f(x) + 2 = (x² + 1) + 2 = x² + 3.
The question involves finding the compositions of two mathematical functions, specifically f ◦ g and g ◦ f. The given functions are f(x) = x² + 1 and g(x) = x + 2. To find f ◦ g, we substitute g(x) into f(x), which results in f(g(x)) = (x + 2)² + 1. Expanding this gives us f(g(x)) = x² + 4x + 5. Similarly, to find g ◦ f, we substitute f(x) into g(x), obtaining g(f(x)) = (x² + 1) + 2, which simplifies to g(f(x)) = x² + 3. These compositions are fundamental operations in algebra that demonstrate how functions can be combined.