Composition of functions

If we have two or more functions that are contained within each other, we call it compound functions. Its general expression is as follows:

\ mathbf {\ left (f \ circ g \ right) (x)) = f (g (x))}

The value of the compound function in x is equal to the function f evaluated in g (x).

function composition - compound functions

Definition

Given two functions f and g, the composite function of f with g is defined as:

\ mathbf {\ left (f \ circ g \ right) (x) = f (g (x))}

his domain is the set of all x (variable) that are in the domain of g, such that g (x) belongs to the domain of f. That is to say:

\ mathbf {Dom (f \ circ g) = \ left \ {x \ in Dom (g) \ left | g (x) \ in Dom (f) \ right \}}

Exercise 1: Sean:

f (x) = 2x 4 -x 2

y

g (x) = \ sqrt {x-4}

Determine:

    1. (f \ circ g)
    2. Your domain

We know that:

(f \ circ g) (x) = f (g (x))

We first substitute the value of g (x):

(f \ circ g) (x) = f (\ sqrt {x-4})

Now, to evaluate f, we substitute x = g (x):

(f \ circ g) (x) = 2 \ sqrt {\ left (x-4 \ right) ^ {4}} - \ sqrt {\ left (x-4 \ right) ^ {2}}

We simplify:

(f \ circ g) (x) = 2 \ (x-4) ^ {2} - (x-4)

We solve the binomial:

\ left (f \ circ g \ right) = 2 \ left (x ^ {2} -8x + 16 \ right) -x + 4

\ left (f \ circ g \ right) = 2x ^ {2} -16x + 32-x + 4

\ left (f \ circ g \ right) = 2x ^ {2} -17x + 36

To determine the domain we must remember that:

Dom (f \ circ g) = \ left \ {x \ in Dom (g) \ left | g (x) \ in Dom (f) \ right \}

We will first look for the domain of g (x); Since the function has a root and the variable is inside it, it cannot be negative:

x-4 \ geq 0

x \ 4

Then the domain of g will be all values greater than or equal to four, that is:

Sun (g) = \ left [4, \ infty \ right)

For the domain of f: since its function is a polynomial, it will be all real numbers, that is:

Sun (f) = \ left (- \ infty, \ infty)

Since we are looking for the domain of the compound function, we know that this domain depends directly on g (x), therefore:

Sun (f \ circ g) = \ left [4, \ infty \ right)