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 fg 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! Keep practicing!