Composite function

A composite function is formed when the output of the first function becomes the input of the second function.

Let f and g be functions and let x be the input of g. Then, g(x)  is the output of function g.

g(x) is the input of function f and the output of function f is f(g(x))

f(g(x) is the composite function of f and g and it is defined as (f ∘ g)(x) = f(g(x))

We can also let x be the input of f. Then, f(x)  is the output of function f.

f(x) is the input of function g and the output of function g is g(f(x))

g(f(x) is a composite function of g and f and it is defined as (g ∘ f)(x) = g(f(x))

Example

Let f(x)  = x + 5 and let g(x) = x² 

Find g ∘ f and f ∘ g

To find g ∘ f, we need to let x be the input of f.

g ∘ f(x) = g(f(x)) = g(x + 5)

Now this is when it is tricky! g(x) = x²

Notice that after we do the composition of functions g and f, x is replaced with x + 5 in g(x). We need to do the same thing with x² and replace x with x + 5

g(f(x))   =     g(x + 5)    =     (x + 5)²

                                     =   x² + 5x + 5x + 25

                                     = x² + 10x + 25

To find f ∘ g, we need to let x be the input of g.

f ∘ g(x) = f(g(x)) = f(x²)

f(x) = x + 5

Notice that after we do the composition of functions f and g, x is replaced with x² in f(x). We need to do the same thing with x + 5 and replace x with x²

f(g(x))   =    f(x²)    =  x²  + 5

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