Composition of Functions 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 fg The composition symbol is: Thus

That's nice! But What Is It?

The easiest way to describe composition is to say it is like substitution. In fact 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 - x Then Means substitute the g function for x in the f function… like this f(x)= 2x + 3 f(g(x) )= 2 g(x) + 3

g(x) = 8 - x f(x)= 2x + 3, and f(x)= 2x + 3 f(g(x) )= 2 g(x) + 3 Now substitute what g equals for g(x) f(8 - x)= 2 (8 - x) + 3 = 16 - 2x + 3 = 19 - 2x So, = 19 - 2x

An interesting fact is that most of the time. Let’s see if this is the case for the previous example.

f(x) = 2x + 3, and g(x) = 8 - x Thus we will substitute f into g. g(x) = 8 - x g(f(x) ) = 8 - f(x) Now substitute what f(x) is: g(2x + 3) = 8 - (2x + 3) = 8 - 2x - 3 = 5 - 2x

Those were easy! My homework is never that easy!

Okay! I'll make it harder. Let and Is that better?

Step 1 Step 2 Step 3 Replace g(x) with Step 4 Simplify Write the f function Step 2 Substitute g(x) for x Step 3 Replace g(x) with Step 4 Simplify

Your Turn! Find: A) B) When ready click your mouse. The answer is: Move your mouse over the correct answer. B)

Your Turn Again! Find: A) B) The answer is: When ready click your mouse. A) Move your mouse over the correct answer. B)

Once More! Come On! Find: A) B) The answer is: When ready click your mouse. A) Move your mouse over the correct answer. B)

Ans. A for the previous example Was actually A)

Practice makes perfect! Do the practice problems following This module in your module section Of Blackboard. Check your work using This live math page.