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Operations with Functions

Operations with Functions. Lesson 2.4. OBJECTIVES: Perform operations with functions to write new functions. Find the composition of two functions. Application: Stopping the car.

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Operations with Functions

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  1. Operations with Functions Lesson 2.4

  2. OBJECTIVES: • Perform operations with functions to write new functions. • Find the composition of two functions.

  3. Application: Stopping the car The faster you drive the longer it takes to stop your car. Total stopping distance is the reaction distance plus the braking distance. Put the data in the table into L1, L2, L3, and L4 and answer the questions on p. 111.

  4. Application: Stopping the car

  5. Hint for entering the speed: Go to List, OPS, seq( Enter x, x, 10, 70, 10) This tells the calculator you would like to enter a sequence of numbers beginning with 10, ending with 70, counting by tens. The home screen will show: {10, 20, 30, …} Store this is L1 by hitting STO, L1.

  6. List 2 can be done the same way with seq(X,X,11,77,11) Enter list 3 by hand What do you notice about Stopping distance? List 4 is the sum of list 2 and list 3. To do this in one step, arrow up so that the cursor is on L4 and type in L1 + L2 and hit ENTER.

  7. Calculate the linear regression for L1 and L2. Calculate the quadratic regression for L1 and L3. Calculate the quadratic regression for L1 and L4. What do you notice about the quadratic regression for L1 and L4?

  8. The linear regression of speed to reaction distance is y = 1.1x The quadratic regression of speed to braking distance is y  .0526x2 + .00x - .29 The quadratic regression of speed to stopping distance is y  .0526x2 + 1.1x - .29 (the sum of the previous two)

  9. Example 1: Let f(x) = 5x2 - 2x + 3 and g(x) = 4x2 + 7x - 5 a. Find (f + g)(x) This means f(x) + g(x) 5x2 - 2x + 3 + (4x2 + 7x - 5) 9x2 + 5x - 2 b. Find (f - g)(x) This means f(x) - g(x). 5x2 - 2x + 3 - (4x2 + 7x - 5) = x2 - 9x + 8

  10. Example 2: Let f(x) = 5x2 and g(x) = 3x - 1 a. Find (f • g)(x) This means f(x) • g(x) (5x2)(3x - 1) (Use the distributive property) =15x3 - 5x2 b. Find (f/g)(x) This means f(x)/g(x). 5x2/(3x - 1) (This cannot be simplified)

  11. Try These Let f(x) = -7x2 + 12x + 2.5 and g(x) = 4x2 + 7x - 5 a. Find (f + g)(x) b. Find (f - g)(x) -3x2 + 19x - 2.5 -11x2 + 5x + 7.5

  12. Try These Let f(x) = 3x2 + 1 and g(x) = 5x - 2 a. Find (f • g)(x) b. Find (f / g)(x) 15x3 - 6x2 + 5x - 2

  13. Homework: p. 115 (11 - 34 , 60,61)

  14. Composition of functions If f and g are functions with appropriate domains and ranges, then the composition of f with g, denoted f o g is given by: In simple terms this means taking the result of one function and performing the other function on that result.

  15. Composition of functions Go play with Geometer’s Sketchpad Dynagraphs Sketchpad, Samples, Sketches, Algebra, Dynagraphs

  16. Example 3 - Composing functions Let f(x) = x2 - 1 and g(x) = 3x a. Find (fog)(x) This means f(g(x)). g(x) = 3x so f(g(x) means f(3x). Since f(x) = x2 - 1 then f(3x) = (3x)2 - 1 or 9x2 - 1

  17. Example 3 - Composing functions Let f(x) = x2 - 1 and g(x) = 3x b. Find (gof)(x) This means g(f(x)). f(x) = x2 - 1 so g(f(x) means g(x2 - 1). Since g(x) = 3x then g(x2 - 1) = 3(x2 - 1) or 3x2 - 3 Notice that (fog)(x) and (gof)(x) are not commutative.

  18. Example 3c - Composing functions Let f(x) = 2x - 1 and g(x) = -4x a. Evaluate (fog)(3) Find g(3) first. g(3) = -4(3) = -12. Then find f(-12) f(-12) = 2(-12) - 1 = -24 - 1 = -25 b. Evaluate (go f)(3) Find f(3) first. f(3) = 2(3) - 1 = 6 - 1 = 5. Then find g(5). g(5) = -4(5) = -20 c. Evaluate (fof)(3). f(3) = 5, f(5) = 9.

  19. Composing functions - Try these Let f(x) = -2x2 + 3 and g(x) = -2x. Find (fog)(x)and (gof)(x) (fog)(x) = -8x2 + 3 (gof)(x) = 4x2 - 6

  20. Composing functions - Challenge Let f(x) = x2, g(x) = 3x, and h(x) = x + 5 1. Find [fo (g o h)](x) 9x2 + 90x + 225 2. Find [go (f o h)](x) 3x2 + 30x + 75 3. Find [ho (g o f)](x) 3x2 + 5

  21. Composing functions application A local electronics store is offering a $100 rebate along with a 10% discount. Let x represent the original price of an item in the store. a. Write the function D that represents the sale price after a 10% discount and the function R that represents the sale price after a $100 rebate.

  22. Composing functions application Since a 10% discount on the original price is the same as paying 90% of the original price, D(x) = x - 0.1x or D(x) = 0.9x. The rebate function is R(x) = x - 100

  23. Composing functions application A local electronics store is offering a $100 rebate along with a 10% discount. Let x represent the original price of an item in the store. b. Find the composite functions (R o D)(x) and (D o R)(x) and explain what they represent. Which is the better deal?

  24. (R o D)(x) is taking the 10% discount first. R(D(x)) = R(0.9x) = (0.9x) - 100 = 0.9x - 100 (D o R)(x) is taking the rebate first. D(R(x)) = D(x - 100) = 0.9(x - 100) 0.9x - 90 The discount first is the better deal.

  25. Homework: p. 115 (35 – 41 odd, 43 – 58, 62 - 64)

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