1. Composition of
Functions
By: Dr. Julia Arnold

2. 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

3. That's nice!
But What Is It?

4. 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.

5. 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

6. 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
= x + 3
= x
So, = x

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

8. 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

9. Those were easy!
My homework is
never that easy!

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

11. 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

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

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

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

15. Ans. A for the previous example
Was actually A)

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