2. Aim: Function Operations Course: Alg. 2 & Trig. Aim: What are some ways that functions can operate? Do Now: Evaluate f(x) = 3x and g(x) = x – 5 if x = 3. f(3) = 3x g(3) = x – 5 f(3) = 3(3) g(3) = (3) – 5 f(3) = 9 g(3) = -2 What is the value of f(3) + g(3)?9 + -2 = 7 What is the sum of f(x) + g(x)?4x – 5 What is the domain for f(x) + g(x)? All real numbers

3. Aim: Function Operations Course: Alg. 2 & Trig. Operation Resulting Function Domain Function Operations Addition f(x) + g(x)All reals f(x) = 3x and g(x) = x – 5 = 3x + (x – 5) = 4x – 5 Subtraction f(x) – g(x)All reals = 3x – (x – 5) = 2x + 5 Multiplication f(x) g(x)All reals = (3x)(x – 5) = 3x 2 – 15x Division f(x)  g(x) All reals except 5

4. Aim: Function Operations Course: Alg. 2 & Trig. f(x) = -3x 2 + 4x + 5 Finding the Domain of a Function All reals All non-negative reals What is the domain for the following functions? What numeric inputs make valid statements? All reals except four Understanding domain will clarify graphing of function.

5. Aim: Function Operations Course: Alg. 2 & Trig. Composition of Functions A binary operation that processes the results of one function through a second function. f(x) = x 2 and g(x) = x + 4 1st input x f(?) = ? 2 x2x2 1st output x2x2 2nd input x2x2 g(?) = ? + 4 x 2 + 4 2nd output x 2 + 4

6. Aim: Function Operations Course: Alg. 2 & Trig. f(?) = ? 2 Composition of Functions f(x) = x 2 and g(x) = x + 4 1st input x 4242 1st output 16 2nd input 16 g(?) = ? + 4 16 + 4 2nd output 20 let x = 4 4Composition

7. Aim: Function Operations Course: Alg. 2 & Trig. Composition of Functions Notation f(x) = x 2 and g(x) = x + 4 let x = 4 f(x)f(x) g( ) How do we symbolically show this composition of functions? 1.Evaluate the inner function f(x) first. 1 2.Then use the answer as the input for the outer function g(x). 2 g(f(4)) = 20

8. Aim: Function Operations Course: Alg. 2 & Trig. Composition of Functions Notation f(x)f(x) 1.Evaluate the inner function f(x) first. g( ) 2.Then use the answer as the input for the outer function g(x). 1 2 A composite function is a new function created by the output of one function used as the input of a second function. g o f g of f of x Read as “ g of f of x”

9. Aim: Function Operations Course: Alg. 2 & Trig. Model Problems Find the composition of f(x) = 4x with g(x) = 2 – x f(g(x)) = f(2 – x) g(x)g(x) Substitute 2 - x for g(x) 4(2 – x)Apply f( ) = 4( ) 8 – 4xSimplify Find the composition of g(x) = 2 – x with f(x) = 4x g(f(x)) = g(4x)g(4x) f(x)f(x) Substitute 4x for f(x) 2 – (4x)Apply g( ) = 2 – ( ) 2 – 4xSimplify Composition of functions are not commutative f(g(x))  g(f(x)) note wording

10. Aim: Function Operations Course: Alg. 2 & Trig. Model Problems Let h(x) = x 2 and r(x) = x + 3 a.Evaluate (h  r)(5) b.Find the rule of the function (h  r)(x) a. r(x) = x + 3 r(5) = 5 + 3 = 8 h(x) = x 2 h(8) = 8 2 = 64 b. (h  r)(x) = h( r(x)) h( x + 3) h(x) = x 2 h( x + 3) = (x + 3) 2 = x 2 + 6x + 9 (h  r)(x) = x 2 + 6x + 9 Evaluate x 2 + 6x + 9 when x = 5

11. Aim: Function Operations Course: Alg. 2 & Trig. Model Problems A store offers a 10% discount sale on its $25 jeans. You also have a coupon worth $5 off any item. Which is a better deal: 10% off first then subtract the $5 or subtract the $5 and then discount the rest at 10%? Use functional notation/operations; let x represent the original price f(x) = x – 5 g(x) =.90x subtract $5 10% discount g(f(x)) subtract $5; discount 10% discount 10%; subtract $5 f(g(x)) g(x – 5) =.90(x – 5)f(.90x) =.90x – 5 g(f(25))f(g(25)) = 18= 17.50

12. Aim: Function Operations Course: Alg. 2 & Trig. Hand-in Assignment The regular price of a certain new car is $15,800. The dealership advertised a factory rebate of $1500 and a 12% discount. Compare the sale price obtained by subtracting the rebate first, then taking the discount, with the sale price obtained by taking the discount first, then subtracting the rebate. Use function notation as in a similar problem in your text book (Book B) on page 28 to explain your resulting conclusions on looseleaf – to be handed in.