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

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

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

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)

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)

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)

Ans. A for the previous example Was actually A)

Together: go over pg. 22-23 (up to example 3) Then solve # 13-18 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

RADIONUCLIDE HALF-LIFE Radon-222 4 days Iodine-131 8 days Krypton-85 10 years Tritium (Hydrogen-3) 12 years Strontium-90 29 years Cesium-137 30 years Americium-241 433 years Radium-226 1,622 years Plutonium-239 24,000 years Uranium-235 is 713,000,000 years Uranium-238 is 4500,000,000 years.

9/25/12 Exponential Functions The number e Objectives: Number e and applications Homework: # 26 pg.27, go over # 41-46 pg.28 quiz Thursday go over AP quiz pg. 29 Do now: # 40 pg 28 (39-was a h/w)

As n increases sequence (1+1/n)n approaches to the number 2.718…

Number e 2.718281828459045235360287471352662497775724709369995…

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