Algebra 1 - Composite Functions

Introduction

  • Composite Function - Given the functions f and g, the composite function is denoted by fgx=fgx
  • Composition of function is like combining two functions to produce a new function using the concept of evaluating function (direct substitution of x-values).
  • The notation fgx is read as "f is composed with g."
  • The notation gfx is read as "g is composed with f."
  • In the composite function fgx=fgx, the inner function or independent function is gx, while the dependent function or outer function is fx.
  • In the case of gfx=gfx, the dependent function is gx, and the independent function is fx.
  • The domain of fgx is the set of all real numbers x in the domain of g wherein gx is the domain of fx.
  • The domain of gfx is the set of all real numbers x in the domain of f wherein fx is the domain of gx.

Solved Examples

Example 1. Given that fx=3x-4 and gx=5x+7, find fgx.

Solution:

The composite function fgx is the same as fgx. The dependent function is f, and the independent function is g.

fgx=f5x+7=35x+7-4

f5x+7=15x+17

 

Example 2. If gx=x2-7x, find ggx.

Solution:

The dependent function and independent function are the same. 

ggx=ggx

gx2-7x=x2-7x2-7x2-7x

gx2-7x=x4-14x3+49x2-7x2+49x

gx2-7x=x4-14x3+42x2+49x

 

Example 3. Given that fx=x2+9, hx=x-4, and gx=5-3x, find hgx+gfx.

Solution:

hgx=hgx=h5-3x=5-3x-4=1-3x

gfx=gfx=gx2+9=5-3x2+9=5-3x2-27=-3x2-22

hgx+gfx=1-3x+-3x2-22=-3x2-3x-21

 

Example 4. If fx=3x2+5 and gx=4x-1, find gf-2.

Solution:

We need to first evaluate fx=3x2+5 at x=-2.

f-2=3-22+5=17

g17=417-1=67

 

Example 5. Given that f=2,3,3,4,4,3 and g=2,4,3,3,4,3, find the ordered pair for the function fgx. What is the domain of fgx?

Solution:

fg2=f4=32,3

fg3=f3=43,4

fg4=f3=44,4

Thus, the composite function fgx is 2,2,3,4,4,4, and its domain is D:2,3,4.

Solved Examples (Decomposition of Functions)

To decompose a function, we need to determine what functions (independent or dependent) compose a given composite function. Then, we have to break the new function into two functions.

 

Example 1. If fx=x+4 and fgx=4x2+x-2, find gx.

Solution:

The unknown here is the inner function gx.

Let the composite function be equal to the dependent function fx=x+4 where x=gx.

fgx=4x2+x-2

4x2+x-2=gx+4

Thus, the independent function is gx=4x2+x-6.

 

Example 2. If fgx=10x2+5x+1 and gx=5x-3, find fx.

Solution:

Make an equation for x in terms of gx.

gx=5x-3x=gx+35

We substitute x=gx+35 to fgx=10x2+5x+1

fgx+35=10gx+352+5gx+35+1

Let x=gx

fx+35=10x+352+5x+35+1fx=10x+3225+x+3+1

fx=2x2+17x+385

Cheat Sheet

  • Evaluating functions is an important skill to master to find the composition of two functions correctly.
  • If fx=x3+2x2-x-1 and gx=x+1, then we find the composite function fgx as fgx=fx+1=x+13+2x+12-x+1-1. Algebraic procedures must be followed to determine the composite function correctly.
  • In example like hx=x and fx=2x+1, the domain of hfx is x-12, and the range of hfx is the range of hx=x.

Blunder Areas

  • In its broad sense, the composition of functions is not commutative. This means that fgx is not always equal to gfx. There are only selected cases where commutativity is applicable.