Functions Operations. To add, subtract, multiply, and divide functions. To find the composite of two functions.

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1 Functions Operations. To add, subtract, multiply, and divide unctions. To ind the composite o two unctions.

quotient unctions consists o the x-values that are in the domains o both and g.

2 Take a note: Key Concepts Addition Subtraction Multiplication Division gx x gx gx x gx gx x gx x x, gx 0 g gx The domains o the sum, dierence, product, and quotient unctions consists o the x-values that are in the domains o both and g. Also, the domain o the quotient unction does not contain any x-value or which g(x) 0.

3 Example #1 Let x 4x 7 and gx x x.what are g and g? What are their domains? Answers gx x gx x 7 x x gx x gx x 7 x x 4 5x x 7 4 3x x 7 The domain o is the set o all real numbers. The domain o g is all x 0. The domain o both +g and -g is the set o numbers common to the domains o both and g, which is all x 0. our turn: x x Let What are their domains? x 8 and g x 3. What are Answers g and -g? gx x x 5,domain all real numbers. gx x x 11,domain all real numbers.

4 Example #: Multiplying and Dividing Functions. Let x x 9 and g x x 3.What are g and and their g domains? Answer gx x gx g x The domain o g g x x 9 x 3 9 x 3 x x 3 3x 9x 7 x x 3x 3 x 3 The domain o both and g is the set o all real numbers, so the domain o D : x 3, x 3 is the set o all real numbers except 3,even though, ater the simpliication is all real numbers Your turn: Let x 3x 11x 4 and gx 3x 1. What are g and and their domains? 3 Answers gx 9x 30x 3x 4,Domain :, g x x 4,Domain : D, :, 3 3, g is also the set o all real numbers g,,.

5 ake a note: The composition o unction g with unction is written as g x g x.the unction composition is not commutative. Evaluate (x) irst. Then use (x) as the input or g. Example #3: Composing Functions. Let Answer x x 5 and gx x.what is g 3? g x g 8 8 g 3 g 3 5 g and is deined as 64 Your turn What is g 3? Answer 4

6 Example 4: Using Composite Functions. You have a coupon good or $5 o the price o any large pizza. You also get a 10% discount on any pizza i you show your student ID. How much more would you pay or a large pizza i the cashier applies the coupon irst? Answer Let x=the price o a large pizza. Cost using the coupon: C(x)=x-5 D x x Cost using the 10% discount: Compose the unctions to apply the discount then the coupon C Dx CDx C 0.9x 0.9x 5 Subtract the unctions: D C x C D x 0.9x x Compose the unctions to apply the coupon and then the discount. D Cx DCx D 0.10x 0.9x x 5 0.9x 5 0.9x 4. 5 You pay $.50 more i the cashier applies the coupon irst.

7 Your turn: A store is oering a 15% discount on all items. Also, employees get a 0% employee discount. Write composite unctions. a) To model taking the 15% discount and then 0% discount. b) To model taking 0% discount and then 15% discount. c) I you were an employee, which discount would you take irst and why? Answer D(x)=cost ater applying 15% store discount. E(x)= cost ater applying the 0% employee discount, and x=cost o items. D x 0.85x and Ex 0.80x ) E Dx EDx x 0. x a 68 D Ex DEx x 0. x b) 68 c )The total discounts are the same.

8 Take a note:

9 Example: The most basic rational unction is Answer Look at the end behavior x 1 x. Graph it. x increases or decreases without a bound, the graph is approaching the line y=0

10 Increasing, Decreasing, and Constant Functions. 1. Increasing :i x x whenever x x or any x any open interval x x x whenever x or any valuesany open interval.. Deceasing :i 1 1 x x x or any values x and in the interval. 3. Constant :i 1 1 x

11 Some books called local maximum or local minimum. Example: Answers

12 Example : Determine whether each o the ollowing unctions is even, odd or neither. 3 3 a) x x 6x Answers a x x 6 x b) g c) h c) h 4 x x x x x x 1 x x x1 x x 1 ) x 3 6x Because (-x)=-(x), is an odd unction. 4 4 b )g x x x x x Because g(-x)=g(x), is an even unction. Because and h h-x hx, h is -x hx, h is not an odd unction. not an even unction

13 Even unctions and y-axis o symmetry. Look at the points (3,5) and (-3,5) Odd unctions and origin symmetry. Look at the points (,8) and (-,-8)

14 Key eatures: Domain: The input or x-values Range: The output or y-values Maximum: is the largest unction value (output/range) Minimum: is the smallest unction value(output/range) End behavior: Arrow notation X-intercepts: where the graph o the unction crosses the X-axis. Y-intercepts: where the graph o the unction crosses the Y-axis. Increasing intervals: Going up when read rom let to right Decreasing intervals: Going down rom let to right. Symmetry: to the y-axis (-x)=-(x) to the origin (-x)=(x)

15 CW odd and HW even

Function Operations and Composition VOCABULARY. Function Operations

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