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 fg
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!
Keep practicing!