By: Dr. Julia Arnold

Composition is a binary operation like addition , subtraction, multiplication and division are binary operations. (meaning they operate on two elements)

f+g

f-g

fg g

f

The composition symbol is: Thus gf

The easiest way to describe composition is to say it is like substitution. In fact

))(()( xgfxgf

Read f of g of x which means substitute g(x) for x in the f(x) expression.

For example:Suppose f(x)= 2x + 3, and

g(x) = 8 - xThen ))(()( xgfxgf

Means substitute the g function forx in the f function… like this

f(x)= 2x + 3

f(g(x) )= 2 g(x) + 3

f(x)= 2x + 3, and g(x) = 8 - x ))(()( xgfxgf

f(x)= 2x + 3

f(g(x) )= 2 g(x) + 3Nowsubstitute what g equals for g(x)

f(8 - x)= 2 (8 - x) + 3

= 16 - 2x + 3= 19 - 2x

So, = 19 - 2x xgf

An interesting fact is that

xfgxgf most of the time.

Let’s see if this is thecase for theprevious example.

f(x) = 2x + 3, andg(x) = 8 - x

))(( xfgxfg Thus we will substitute f into g.

g(x) = 8 - x

g(f(x) ) = 8 - f(x)

Nowsubstitutewhat f(x) is:

g(2x + 3) = 8 - (2x + 3)

= 8 - 2x - 3= 5 - 2x

Let 32)( 2 xxxfand

xxg )(

32)( 2 xxxf xxg )(

))(()( xgfxgf

Write the f function 32)( 2 xxxfSubstitute g(x) for x 3)(2)())(( 2 xgxgxgf

Replace g(x) with x 32)(2

xxxf

Simplify 32))(( xxxgf

Step 1

Step 2

Step 3

Step 4

32)( 2 xxxf xxg )(

Find: xfg When ready click your mouse.

The answer is:

A) 322 xx

B) 32 xxMove your mouse overthe correct answer.

xxxf 2)(xx

xg

2

1)(

Find: xgf When ready click your mouse.

The answer is:

B) xxxx

2

2

2

11

A)xxxx

224

11

Move your mouse overthe correct answer.

1)( xxf 21

)( x

xg

Find: xfg When ready click your mouse.

The answer is:

B) 2

1

1

x

A) 121

121

x

x

xMove your mouse overthe correct answer.

1)( xxf 21

)( x

xg

A) 121

121

x

x

x

Ans. A for the previous example

Was actually xgf

Do the practice problems followingThis module in your module section Of Blackboard. Check your work usingThis live math page.