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

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

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

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

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

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

15. Together: go over pg. 22-23 (up to example 3)
Then solve # pg. 26 # 1-11 odd pg. 26
To find ZERO: Calculator: y=5-2.5^x→Window (you can set appropriate window x: (-3,3) y: (-6,6) →SECOND TRACE(CALC) →2(ZERO) → “left bound?-chose number close to intersection from left-move cursor to 1.65→ENTER→”right bound?” chose number close to intersection from the right-move cursor to x=1.87→ENTER,ENTER

17. RADIONUCLIDE HALF-LIFE
Radon days
Iodine days
Krypton years
Tritium (Hydrogen-3) 12 years
Strontium years
Cesium years
Americium years
Radium ,622 years
Plutonium ,000 years
Uranium-235 is 713,000,000 years
Uranium-238 is 4500,000,000 years.

18. 9/25/12 Exponential Functions The number e
Objectives: Number e and applications
Homework: # 26 pg.27, go over # pg.28 quiz Thursday
go over AP quiz pg. 29
Do now: # 40 pg 28 (39-was a h/w)

20. As n increases sequence (1+1/n)n
approaches to the number …

21. Number e …

22. Number e is named after Euler
(1+1/n)n