Composite Function Ordering (g(f(x))

Let's try the reverse of f(g(x)) or g(f(x)). Because the g(x) function is now written before (or outside) the f(x) function, the expression g(f(x)) means to substitute f(x) into the g(x) function

For example, if f(x) = 3x+1 and g(x) = 5x², then g(f(x)) would be:

g(f(x)) = 5•(g(x))²
            = 5•(3x + 1)²
            = 5•(9x²+ 6x + 1)
            = 45x²+ 30x + 5

By reversing the order in which the functions are sequentially processed, we get two different results.

f(g(x)) = 15x² + 1
g(f(x)) = 45x²+ 30x + 5

When evaluating functions, it is important to substitute functions in the order specified.

Note: The notation (f ° g)(x) and (g ° f)(x) are both equivalent to f(g(x)) and g(f(x)) respectively.