Functions

Composite Functions

Form (f∘g)(x)=f(g(x)). With f(x)=2x+1 and g(x)=x², substituting g into f gives 2x²+1. At x=3: 2(9)+1=19.

composite-functions To compute f∘g, replace every x in f(x) with the entire expression g(x). Evaluate by substituting the x value.

Example

Evaluate the inner function first, then feed that result into the outer function.

highlighted = computed this step

Step 1 — Set up

Set up the expression.

f(x)=2x+1,g(x)=x2f(x)= 2x + 1 ,\quad g(x)= x^{2}

Step 2 — Evaluate inner function

Evaluate the inner function first: 3^2 = 9.

g(3)=32=9g( 3 )= 3 ^{ 2 }= \hl{9}

Step 3 — Substitute into outer function

Now substitute g(3) = 9 into f.

(f∘g)(3)=f(9)(f\circ g)( 3 )=f( \hl{9} )

Step 4 — Evaluate outer function

Evaluate the outer function: 2 times 9 plus 1 = 19.

f(9)=2⋅9+1=19f( 9 )= 2 \cdot 9 + 1 = \hl{19}

Step 5 — Composite result

The composite value is (f after g)(3) = 19.

(f∘g)(3)=19(f\circ g)( 3 )= \hl{19}