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# 1.5 notation and algebra of functions

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### 1.5 notation and algebra of functions

1. 1. Notation and Algebra of Functions
2. 2. Functions are procedures that assign a unique output to each (valid) input. Notation and Algebra of Functions
3. 3. Functions are procedures that assign a unique output to each (valid) input. Most mathematics functions are given in mathematics formulas such as f (x) = x 2 β 2 x + 3 = y . Notation and Algebra of Functions
4. 4. Functions are procedures that assign a unique output to each (valid) input. Most mathematics functions are given in mathematics formulas such as f (x) = x 2 β 2 x + 3 = y . Notation and Algebra of Functions name of the function
5. 5. Functions are procedures that assign a unique output to each (valid) input. Most mathematics functions are given in mathematics formulas such as f (x) = x 2 β 2 x + 3 = y . Notation and Algebra of Functions name of the function the input box
6. 6. Functions are procedures that assign a unique output to each (valid) input. Most mathematics functions are given in mathematics formulas such as f (x) = x 2 β 2 x + 3 = y . Notation and Algebra of Functions name of the function the input box the defining formula
7. 7. Functions are procedures that assign a unique output to each (valid) input. Most mathematics functions are given in mathematics formulas such as f (x) = x 2 β 2 x + 3 = y . Notation and Algebra of Functions name of the function the input box the defining formula the output
8. 8. Functions are procedures that assign a unique output to each (valid) input. Most mathematics functions are given in mathematics formulas such as f (x) = x 2 β 2 x + 3 = y . Notation and Algebra of Functions The input box β ( ) β holds the input to be evaluated by the formula.
9. 9. Functions are procedures that assign a unique output to each (valid) input. Most mathematics functions are given in mathematics formulas such as f (x) = x 2 β 2 x + 3 = y . Notation and Algebra of Functions The input box β ( ) β holds the input to be evaluated by the formula. Hence f (2) means to replace x by the input (2) in the formula,
10. 10. Functions are procedures that assign a unique output to each (valid) input. Most mathematics functions are given in mathematics formulas such as f (x) = x 2 β 2 x + 3 = y . Notation and Algebra of Functions The input box β ( ) β holds the input to be evaluated by the formula. Hence f (2) means to replace x by the input (2) in the formula, so f (2) = (2) 2 β 2 (2) + 3
11. 11. Functions are procedures that assign a unique output to each (valid) input. Most mathematics functions are given in mathematics formulas such as f (x) = x 2 β 2 x + 3 = y . Notation and Algebra of Functions The input box β ( ) β holds the input to be evaluated by the formula. Hence f (2) means to replace x by the input (2) in the formula, so f (2) = (2) 2 β 2 (2) + 3 = 3 = y.
12. 12. Functions are procedures that assign a unique output to each (valid) input. Most mathematics functions are given in mathematics formulas such as f (x) = x 2 β 2 x + 3 = y . Notation and Algebra of Functions The input may be other mathematics expressions. The input box β ( ) β holds the input to be evaluated by the formula. Hence f (2) means to replace x by the input (2) in the formula, so f (2) = (2) 2 β 2 (2) + 3 = 3 = y.
13. 13. Functions are procedures that assign a unique output to each (valid) input. Most mathematics functions are given in mathematics formulas such as f (x) = x 2 β 2 x + 3 = y . Notation and Algebra of Functions The input may be other mathematics expressions. Often in such problems we are to simplify the outputs algebraically. The input box β ( ) β holds the input to be evaluated by the formula. Hence f (2) means to replace x by the input (2) in the formula, so f (2) = (2) 2 β 2 (2) + 3 = 3 = y.
14. 14. Functions are procedures that assign a unique output to each (valid) input. Most mathematics functions are given in mathematics formulas such as f (x) = x 2 β 2 x + 3 = y . Notation and Algebra of Functions The input may be other mathematics expressions. Often in such problems we are to simplify the outputs algebraically. The input box β ( ) β holds the input to be evaluated by the formula. Hence f (2) means to replace x by the input (2) in the formula, so f (2) = (2) 2 β 2 (2) + 3 = 3 = y. (The SquareβFormula) (a Β± b) 2 = a 2 Β± 2ab + b 2 We write down the SquareβFormula as a reminder below.
15. 15. Functions are procedures that assign a unique output to each (valid) input. Most mathematics functions are given in mathematics formulas such as f (x) = x 2 β 2 x + 3 = y . Notation and Algebra of Functions The input may be other mathematics expressions. Often in such problems we are to simplify the outputs algebraically. The input box β ( ) β holds the input to be evaluated by the formula. Hence f (2) means to replace x by the input (2) in the formula, so f (2) = (2) 2 β 2 (2) + 3 = 3 = y. (The SquareβFormula) (a Β± b) 2 = a 2 Β± 2ab + b 2 We write down the SquareβFormula as a reminder below. Example A. Given f(x) = x 2 β 2x + 3, simplify the following. a. f (2a) = b. f (a + b) =
16. 16. Functions are procedures that assign a unique output to each (valid) input. Most mathematics functions are given in mathematics formulas such as f (x) = x 2 β 2 x + 3 = y . Notation and Algebra of Functions The input may be other mathematics expressions. Often in such problems we are to simplify the outputs algebraically. The input box β ( ) β holds the input to be evaluated by the formula. Hence f (2) means to replace x by the input (2) in the formula, so f (2) = (2) 2 β 2 (2) + 3 = 3 = y. (The SquareβFormula) (a Β± b) 2 = a 2 Β± 2ab + b 2 We write down the SquareβFormula as a reminder below. Example A. Given f(x) = x 2 β 2x + 3, simplify the following. a. f (2a) = (2a) 2 β 2 (2a) + 3 b. f (a + b) =
17. 17. Functions are procedures that assign a unique output to each (valid) input. Most mathematics functions are given in mathematics formulas such as f (x) = x 2 β 2 x + 3 = y . Notation and Algebra of Functions The input may be other mathematics expressions. Often in such problems we are to simplify the outputs algebraically. The input box β ( ) β holds the input to be evaluated by the formula. Hence f (2) means to replace x by the input (2) in the formula, so f (2) = (2) 2 β 2 (2) + 3 = 3 = y. (The SquareβFormula) (a Β± b) 2 = a 2 Β± 2ab + b 2 We write down the SquareβFormula as a reminder below. Example A. Given f(x) = x 2 β 2x + 3, simplify the following. a. f (2a) = (2a) 2 β 2 (2a) + 3 = 4a 2 β 4a + 3 b. f (a + b) =
18. 18. Functions are procedures that assign a unique output to each (valid) input. Most mathematics functions are given in mathematics formulas such as f (x) = x 2 β 2 x + 3 = y . Notation and Algebra of Functions The input may be other mathematics expressions. Often in such problems we are to simplify the outputs algebraically. The input box β ( ) β holds the input to be evaluated by the formula. Hence f (2) means to replace x by the input (2) in the formula, so f (2) = (2) 2 β 2 (2) + 3 = 3 = y. (The SquareβFormula) (a Β± b) 2 = a 2 Β± 2ab + b 2 We write down the SquareβFormula as a reminder below. Example A. Given f(x) = x 2 β 2x + 3, simplify the following. a. f (2a) = (2a) 2 β 2 (2a) + 3 = 4a 2 β 4a + 3 b. f (a + b) = (a + b) 2 β 2 (a + b) + 3
19. 19. Functions are procedures that assign a unique output to each (valid) input. Most mathematics functions are given in mathematics formulas such as f (x) = x 2 β 2 x + 3 = y . Notation and Algebra of Functions The input may be other mathematics expressions. Often in such problems we are to simplify the outputs algebraically. The input box β ( ) β holds the input to be evaluated by the formula. Hence f (2) means to replace x by the input (2) in the formula, so f (2) = (2) 2 β 2 (2) + 3 = 3 = y. (The SquareβFormula) (a Β± b) 2 = a 2 Β± 2ab + b 2 We write down the SquareβFormula as a reminder below. Example A. Given f(x) = x 2 β 2x + 3, simplify the following. a. f (2a) = (2a) 2 β 2 (2a) + 3 = 4a 2 β 4a + 3 b. f (a + b) = (a + b) 2 β 2 (a + b) + 3 = a 2 + 2ab + b 2 β 2a β 2b + 3
20. 20. We may form new functions by +, β , * , and / functions. Notation and Algebra of Functions
21. 21. We may form new functions by +, β , * , and / functions. Notation and Algebra of Functions Example B. Let f(x) = 3x β 4, g(x) = x 2 β 2x β 3, find f + g, f β g, f * g, and f/g.
22. 22. We may form new functions by +, β , * , and / functions. Notation and Algebra of Functions Example B. Let f(x) = 3x β 4, g(x) = x 2 β 2x β 3, find f + g, f β g, f * g, and f/g. f + g = 3x β 4 + x 2 β 2x β 3
23. 23. We may form new functions by +, β , * , and / functions. Notation and Algebra of Functions Example B. Let f(x) = 3x β 4, g(x) = x 2 β 2x β 3, find f + g, f β g, f * g, and f/g. f + g = 3x β 4 + x 2 β 2x β 3 = x 2 + x β 7
24. 24. We may form new functions by +, β , * , and / functions. Notation and Algebra of Functions Example B. Let f(x) = 3x β 4, g(x) = x 2 β 2x β 3, find f + g, f β g, f * g, and f/g. f + g = 3x β 4 + x 2 β 2x β 3 = x 2 + x β 7 f β g = 3x β 4 β (x 2 β 2x β 3)
25. 25. We may form new functions by +, β , * , and / functions. Notation and Algebra of Functions Example B. Let f(x) = 3x β 4, g(x) = x 2 β 2x β 3, find f + g, f β g, f * g, and f/g. f + g = 3x β 4 + x 2 β 2x β 3 = x 2 + x β 7 f β g = 3x β 4 β (x 2 β 2x β 3) = 3x β 4 β x 2 + 2x + 3 = βx 2 + 5x β 1
26. 26. We may form new functions by +, β , * , and / functions. Notation and Algebra of Functions Example B. Let f(x) = 3x β 4, g(x) = x 2 β 2x β 3, find f + g, f β g, f * g, and f/g. f + g = 3x β 4 + x 2 β 2x β 3 = x 2 + x β 7 f β g = 3x β 4 β (x 2 β 2x β 3) = 3x β 4 β x 2 + 2x + 3 = βx 2 + 5x β 1 f * g = (3x β 4) * (x 2 β 2x β 3)
27. 27. We may form new functions by +, β , * , and / functions. Notation and Algebra of Functions Example B. Let f(x) = 3x β 4, g(x) = x 2 β 2x β 3, find f + g, f β g, f * g, and f/g. f + g = 3x β 4 + x 2 β 2x β 3 = x 2 + x β 7 f β g = 3x β 4 β (x 2 β 2x β 3) = 3x β 4 β x 2 + 2x + 3 = βx 2 + 5x β 1 f * g = (3x β 4) * (x 2 β 2x β 3) = 3x 3 β 10x 2 β x + 12
28. 28. We may form new functions by +, β , * , and / functions. Notation and Algebra of Functions Example B. Let f(x) = 3x β 4, g(x) = x 2 β 2x β 3, find f + g, f β g, f * g, and f/g. f + g = 3x β 4 + x 2 β 2x β 3 = x 2 + x β 7 f β g = 3x β 4 β (x 2 β 2x β 3) = 3x β 4 β x 2 + 2x + 3 = βx 2 + 5x β 1 f * g = (3x β 4) * (x 2 β 2x β 3) = 3x 3 β 10x 2 β x + 12 f/g = (3x β 4)/(x 2 β 2x β 3)
29. 29. We may form new functions by +, β , * , and / functions. Notation and Algebra of Functions Example B. Let f(x) = 3x β 4, g(x) = x 2 β 2x β 3, find f + g, f β g, f * g, and f/g. f + g = 3x β 4 + x 2 β 2x β 3 = x 2 + x β 7 f β g = 3x β 4 β (x 2 β 2x β 3) = 3x β 4 β x 2 + 2x + 3 = βx 2 + 5x β 1 f * g = (3x β 4) * (x 2 β 2x β 3) = 3x 3 β 10x 2 β x + 12 f/g = (3x β 4)/(x 2 β 2x β 3) Algebraic expressions may be formed with functions notations.
30. 30. We may form new functions by +, β , * , and / functions. Notation and Algebra of Functions Example B. Let f(x) = 3x β 4, g(x) = x 2 β 2x β 3, find f + g, f β g, f * g, and f/g. f + g = 3x β 4 + x 2 β 2x β 3 = x 2 + x β 7 f β g = 3x β 4 β (x 2 β 2x β 3) = 3x β 4 β x 2 + 2x + 3 = βx 2 + 5x β 1 f * g = (3x β 4) * (x 2 β 2x β 3) = 3x 3 β 10x 2 β x + 12 f/g = (3x β 4)/(x 2 β 2x β 3) Algebraic expressions may be formed with functions notations. Example C. Let f(x) = x 2 β 2x β 3, g(x) = 2x + 1. a. Simplify f(2) β g(β3)
31. 31. We may form new functions by +, β , * , and / functions. Notation and Algebra of Functions Example B. Let f(x) = 3x β 4, g(x) = x 2 β 2x β 3, find f + g, f β g, f * g, and f/g. f + g = 3x β 4 + x 2 β 2x β 3 = x 2 + x β 7 f β g = 3x β 4 β (x 2 β 2x β 3) = 3x β 4 β x 2 + 2x + 3 = βx 2 + 5x β 1 f * g = (3x β 4) * (x 2 β 2x β 3) = 3x 3 β 10x 2 β x + 12 f/g = (3x β 4)/(x 2 β 2x β 3) Algebraic expressions may be formed with functions notations. Example C. Let f(x) = x 2 β 2x β 3, g(x) = 2x + 1. a. Simplify f(2) β g(β3) We find f(2) = (2) 2 β 2 * (2) β 3
32. 32. We may form new functions by +, β , * , and / functions. Notation and Algebra of Functions Example B. Let f(x) = 3x β 4, g(x) = x 2 β 2x β 3, find f + g, f β g, f * g, and f/g. f + g = 3x β 4 + x 2 β 2x β 3 = x 2 + x β 7 f β g = 3x β 4 β (x 2 β 2x β 3) = 3x β 4 β x 2 + 2x + 3 = βx 2 + 5x β 1 f * g = (3x β 4) * (x 2 β 2x β 3) = 3x 3 β 10x 2 β x + 12 f/g = (3x β 4)/(x 2 β 2x β 3) Algebraic expressions may be formed with functions notations. Example C. Let f(x) = x 2 β 2x β 3, g(x) = 2x + 1. a. Simplify f(2) β g(β3) We find f(2) = (2) 2 β 2 * (2) β 3 = 4 β 4 β 3
33. 33. We may form new functions by +, β , * , and / functions. Notation and Algebra of Functions Example B. Let f(x) = 3x β 4, g(x) = x 2 β 2x β 3, find f + g, f β g, f * g, and f/g. f + g = 3x β 4 + x 2 β 2x β 3 = x 2 + x β 7 f β g = 3x β 4 β (x 2 β 2x β 3) = 3x β 4 β x 2 + 2x + 3 = βx 2 + 5x β 1 f * g = (3x β 4) * (x 2 β 2x β 3) = 3x 3 β 10x 2 β x + 12 f/g = (3x β 4)/(x 2 β 2x β 3) Algebraic expressions may be formed with functions notations. Example C. Let f(x) = x 2 β 2x β 3, g(x) = 2x + 1. a. Simplify f(2) β g(β3) We find f(2) = (2) 2 β 2 * (2) β 3 = 4 β 4 β 3 = β 3
34. 34. We may form new functions by +, β , * , and / functions. Notation and Algebra of Functions Example B. Let f(x) = 3x β 4, g(x) = x 2 β 2x β 3, find f + g, f β g, f * g, and f/g. f + g = 3x β 4 + x 2 β 2x β 3 = x 2 + x β 7 f β g = 3x β 4 β (x 2 β 2x β 3) = 3x β 4 β x 2 + 2x + 3 = βx 2 + 5x β 1 f * g = (3x β 4) * (x 2 β 2x β 3) = 3x 3 β 10x 2 β x + 12 f/g = (3x β 4)/(x 2 β 2x β 3) Algebraic expressions may be formed with functions notations. Example C. Let f(x) = x 2 β 2x β 3, g(x) = 2x + 1. a. Simplify f(2) β g(β3) We find f(2) = (2) 2 β 2 * (2) β 3 = 4 β 4 β 3 = β 3 g(β3) = 2(β3) +1
35. 35. We may form new functions by +, β , * , and / functions. Notation and Algebra of Functions Example B. Let f(x) = 3x β 4, g(x) = x 2 β 2x β 3, find f + g, f β g, f * g, and f/g. f + g = 3x β 4 + x 2 β 2x β 3 = x 2 + x β 7 f β g = 3x β 4 β (x 2 β 2x β 3) = 3x β 4 β x 2 + 2x + 3 = βx 2 + 5x β 1 f * g = (3x β 4) * (x 2 β 2x β 3) = 3x 3 β 10x 2 β x + 12 f/g = (3x β 4)/(x 2 β 2x β 3) Algebraic expressions may be formed with functions notations. Example C. Let f(x) = x 2 β 2x β 3, g(x) = 2x + 1. a. Simplify f(2) β g(β3) We find f(2) = (2) 2 β 2 * (2) β 3 = 4 β 4 β 3 = β 3 g(β3) = 2(β3) +1 = β6 + 1 = β5
36. 36. We may form new functions by +, β , * , and / functions. Notation and Algebra of Functions Example B. Let f(x) = 3x β 4, g(x) = x 2 β 2x β 3, find f + g, f β g, f * g, and f/g. f + g = 3x β 4 + x 2 β 2x β 3 = x 2 + x β 7 f β g = 3x β 4 β (x 2 β 2x β 3) = 3x β 4 β x 2 + 2x + 3 = βx 2 + 5x β 1 f * g = (3x β 4) * (x 2 β 2x β 3) = 3x 3 β 10x 2 β x + 12 f/g = (3x β 4)/(x 2 β 2x β 3) Algebraic expressions may be formed with functions notations. Example C. Let f(x) = x 2 β 2x β 3, g(x) = 2x + 1. a. Simplify f(2) β g(β3) We find f(2) = (2) 2 β 2 * (2) β 3 = 4 β 4 β 3 = β 3 Hence f(2) β g(-3) = β 3 β (β5) g(β3) = 2(β3) +1 = β6 + 1 = β5
37. 37. We may form new functions by +, β , * , and / functions. Notation and Algebra of Functions Example B. Let f(x) = 3x β 4, g(x) = x 2 β 2x β 3, find f + g, f β g, f * g, and f/g. f + g = 3x β 4 + x 2 β 2x β 3 = x 2 + x β 7 f β g = 3x β 4 β (x 2 β 2x β 3) = 3x β 4 β x 2 + 2x + 3 = βx 2 + 5x β 1 f * g = (3x β 4) * (x 2 β 2x β 3) = 3x 3 β 10x 2 β x + 12 f/g = (3x β 4)/(x 2 β 2x β 3) Algebraic expressions may be formed with functions notations. Example C. Let f(x) = x 2 β 2x β 3, g(x) = 2x + 1. a. Simplify f(2) β g(β3) We find f(2) = (2) 2 β 2 * (2) β 3 = 4 β 4 β 3 = β 3 Hence f(2) β g(-3) = β 3 β (β5) = β3 + 5 = 2 g(β3) = 2(β3) +1 = β6 + 1 = β5
38. 38. b. Simplify f(x + h ) β f(x) Notation and Algebra of Functions
39. 39. b. Simplify f(x + h ) β f(x) First, we calculate f(x + h ) Notation and Algebra of Functions
40. 40. b. Simplify f(x + h ) β f(x) First, we calculate f (x + h ) = (x+h) 2 β 2 (x+h) β 3 Notation and Algebra of Functions
41. 41. b. Simplify f(x + h ) β f(x) First, we calculate f(x + h ) = (x+h) 2 β 2(x+h) β 3 = x 2 + 2xh + h 2 β 2x β 2h β 3 Notation and Algebra of Functions
42. 42. b. Simplify f(x + h ) β f(x) First, we calculate f(x + h ) = (x+h) 2 β 2(x+h) β 3 = x 2 + 2xh + h 2 β 2x β 2h β 3 Hence f(x+h) β f(x) = x 2 + 2xh + h 2 β 2x β 2h β 3 β (x 2 β 2x β 3) Notation and Algebra of Functions
43. 43. b. Simplify f(x + h ) β f(x) First, we calculate f(x + h ) = (x+h) 2 β 2(x+h) β 3 = x 2 + 2xh + h 2 β 2x β 2h β 3 Hence f(x+h) β f(x) = x 2 + 2xh + h 2 β 2x β 2h β 3 β (x 2 β 2x β 3) = x 2 + 2xh + h 2 β 2x β 2h β 3 β x 2 + 2x + 3 Notation and Algebra of Functions
44. 44. b. Simplify f(x + h ) β f(x) First, we calculate f(x + h ) = (x+h) 2 β 2(x+h) β 3 = x 2 + 2xh + h 2 β 2x β 2h β 3 Hence f(x+h) β f(x) = x 2 + 2xh + h 2 β 2x β 2h β 3 β (x 2 β 2x β 3) = x 2 + 2xh + h 2 β 2x β 2h β 3 β x 2 + 2x + 3 = 2xh + h 2 β 2h Notation and Algebra of Functions
45. 45. b. Simplify f(x + h ) β f(x) First, we calculate f(x + h ) = (x+h) 2 β 2(x+h) β 3 = x 2 + 2xh + h 2 β 2x β 2h β 3 Hence f(x+h) β f(x) = x 2 + 2xh + h 2 β 2x β 2h β 3 β (x 2 β 2x β 3) = x 2 + 2xh + h 2 β 2x β 2h β 3 β x 2 + 2x + 3 = 2xh + h 2 β 2h Notation and Algebra of Functions Composition of Functions
46. 46. b. Simplify f(x + h ) β f(x) First, we calculate f(x + h ) = (x+h) 2 β 2(x+h) β 3 = x 2 + 2xh + h 2 β 2x β 2h β 3 Hence f(x+h) β f(x) = x 2 + 2xh + h 2 β 2x β 2h β 3 β (x 2 β 2x β 3) = x 2 + 2xh + h 2 β 2x β 2h β 3 β x 2 + 2x + 3 = 2xh + h 2 β 2h Notation and Algebra of Functions Composition of Functions When one function is used as the input of another function, this operation is called composition .
47. 47. b. Simplify f(x + h ) β f(x) First, we calculate f(x + h ) = (x+h) 2 β 2(x+h) β 3 = x 2 + 2xh + h 2 β 2x β 2h β 3 Hence f(x+h) β f(x) = x 2 + 2xh + h 2 β 2x β 2h β 3 β (x 2 β 2x β 3) = x 2 + 2xh + h 2 β 2x β 2h β 3 β x 2 + 2x + 3 = 2xh + h 2 β 2h Notation and Algebra of Functions Composition of Functions When one function is used as the input of another function, this operation is called composition . Given f(x) and g(x), we define (f β g) (x) β‘ f ( g(x) ).
48. 48. b. Simplify f(x + h ) β f(x) First, we calculate f(x + h ) = (x+h) 2 β 2(x+h) β 3 = x 2 + 2xh + h 2 β 2x β 2h β 3 Hence f(x+h) β f(x) = x 2 + 2xh + h 2 β 2x β 2h β 3 β (x 2 β 2x β 3) = x 2 + 2xh + h 2 β 2x β 2h β 3 β x 2 + 2x + 3 = 2xh + h 2 β 2h Notation and Algebra of Functions Composition of Functions When one function is used as the input of another function, this operation is called composition . Given f(x) and g(x), we define (f β g) (x) β‘ f ( g(x) ). (It's read as f &quot;circled&quot; g).
49. 49. b. Simplify f(x + h ) β f(x) First, we calculate f(x + h ) = (x+h) 2 β 2(x+h) β 3 = x 2 + 2xh + h 2 β 2x β 2h β 3 Hence f(x+h) β f(x) = x 2 + 2xh + h 2 β 2x β 2h β 3 β (x 2 β 2x β 3) = x 2 + 2xh + h 2 β 2x β 2h β 3 β x 2 + 2x + 3 = 2xh + h 2 β 2h Notation and Algebra of Functions Composition of Functions When one function is used as the input of another function, this operation is called composition . Given f(x) and g(x), we define (f β g) (x) β‘ f ( g(x) ). (It's read as f &quot;circled&quot; g). In other words, the input for f is g(x) .
50. 50. b. Simplify f(x + h ) β f(x) First, we calculate f(x + h ) = (x+h) 2 β 2(x+h) β 3 = x 2 + 2xh + h 2 β 2x β 2h β 3 Hence f(x+h) β f(x) = x 2 + 2xh + h 2 β 2x β 2h β 3 β (x 2 β 2x β 3) = x 2 + 2xh + h 2 β 2x β 2h β 3 β x 2 + 2x + 3 = 2xh + h 2 β 2h Notation and Algebra of Functions Composition of Functions When one function is used as the input of another function, this operation is called composition . Given f(x) and g(x), we define (f β g) (x) β‘ f ( g(x) ). (It's read as f &quot;circled&quot; g). In other words, the input for f is g(x) . Similarly, we define (g β f) (x) β‘ g ( f(x) ).
51. 51. Example C. Let f(x) = 3x β 5, g(x) = β4x + 3, simplify (f β g)(3), (g β f)(3). Notation and Algebra of Functions
52. 52. Example C. Let f(x) = 3x β 5, g(x) = β4x + 3, simplify (f β g)(3), (g β f)(3). (f β g)(3) = f( g(3) ) Notation and Algebra of Functions
53. 53. Example C. Let f(x) = 3x β 5, g(x) = β4x + 3, simplify (f β g)(3), (g β f)(3). (f β g)(3) = f( g(3) ) Since g(3) = β4(3) + 3 Notation and Algebra of Functions
54. 54. Example C. Let f(x) = 3x β 5, g(x) = β4x + 3, simplify (f β g)(3), (g β f)(3). (f β g)(3) = f( g(3) ) Since g(3) = β4(3) + 3 = β9 Notation and Algebra of Functions
55. 55. Example C. Let f(x) = 3x β 5, g(x) = β4x + 3, simplify (f β g)(3), (g β f)(3). (f β g)(3) = f( g(3) ) Since g(3) = β4(3) + 3 = β9 Hence f( g(3) ) = f (β9) Notation and Algebra of Functions
56. 56. Example C. Let f(x) = 3x β 5, g(x) = β4x + 3, simplify (f β g)(3), (g β f)(3). (f β g)(3) = f( g(3) ) Since g(3) = β4(3) + 3 = β9 Hence f( g(3) ) = f (β9) = 3 (β9) β 5 = β32 Notation and Algebra of Functions
57. 57. Example C. Let f(x) = 3x β 5, g(x) = β4x + 3, simplify (f β g)(3), (g β f)(3). (f β g)(3) = f( g(3) ) Since g(3) = β4(3) + 3 = β9 Hence f( g(3) ) = f (β9) = 3 (β9) β 5 = β32 (g β f)(3) = g( f(3) ) Notation and Algebra of Functions
58. 58. Example C. Let f(x) = 3x β 5, g(x) = β4x + 3, simplify (f β g)(3), (g β f)(3). (f β g)(3) = f( g(3) ) Since g(3) = β4(3) + 3 = β9 Hence f( g(3) ) = f (β9) = 3 (β9) β 5 = β32 (g β f)(3) = g( f(3) ) Since f(3) = 3(3) β 5 Notation and Algebra of Functions
59. 59. Example C. Let f(x) = 3x β 5, g(x) = β4x + 3, simplify (f β g)(3), (g β f)(3). (f β g)(3) = f( g(3) ) Since g(3) = β4(3) + 3 = β9 Hence f( g(3) ) = f (β9) = 3 (β9) β 5 = β32 (g β f)(3) = g( f(3) ) Since f(3) = 3(3) β 5 = 4 Notation and Algebra of Functions
60. 60. Example C. Let f(x) = 3x β 5, g(x) = β4x + 3, simplify (f β g)(3), (g β f)(3). (f β g)(3) = f( g(3) ) Since g(3) = β4(3) + 3 = β9 Hence f( g(3) ) = f (β9) = 3 (β9) β 5 = β32 (g β f)(3) = g( f(3) ) Since f(3) = 3(3) β 5 = 4 Hence g( f(3) ) =g (4) Notation and Algebra of Functions
61. 61. Example C. Let f(x) = 3x β 5, g(x) = β4x + 3, simplify (f β g)(3), (g β f)(3). (f β g)(3) = f( g(3) ) Since g(3) = β4(3) + 3 = β9 Hence f( g(3) ) = f (β9) = 3 (β9) β 5 = β32 (g β f)(3) = g( f(3) ) Since f(3) = 3(3) β 5 = 4 Hence g( f(3) ) =g (4) = β4(4) + 3 Notation and Algebra of Functions
62. 62. Example C. Let f(x) = 3x β 5, g(x) = β4x + 3, simplify (f β g)(3), (g β f)(3). (f β g)(3) = f( g(3) ) Since g(3) = β4(3) + 3 = β9 Hence f( g(3) ) = f (β9) = 3 (β9) β 5 = β32 (g β f)(3) = g( f(3) ) Since f(3) = 3(3) β 5 = 4 Hence g( f(3) ) =g (4) = β4(4) + 3 = β13 Notation and Algebra of Functions
63. 63. Note: They are different! Example C. Let f(x) = 3x β 5, g(x) = β4x + 3, simplify (f β g)(3), (g β f)(3). (f β g)(3) = f( g(3) ) Since g(3) = β4(3) + 3 = β9 Hence f( g(3) ) = f (β9) = 3 (β9) β 5 = β32 (g β f)(3) = g( f(3) ) Since f(3) = 3(3) β 5 = 4 Hence g( f(3) ) =g (4) = β4(4) + 3 = β13 Notation and Algebra of Functions
64. 64. f(x) = 2 β 3x g(x) = β2x 2 + 3x β 1 h(x) = Exercise A. Simplify the following expressions with the given functions. Notation and Algebra of Functions 2x β 1 x β 2 8. f(2a) 12. 2h(a) 9. g(2a) 11. h(2a) 10. 2g(a) 13. f(3 + b) 14. g(3 + b) 15. h(3 + b) 1. f(2) + f(3) 5. f(0) + g(0) + h(0) 2. 2f(3) 4. [h(2)] 2 3. 2g(0) + g(1) 6. [f(3)] 2 β [g(3)] 2 7. h(1) / h(β1 ) 16. f(3 + b) β f(b) 17. g(3 + b) β g(b) 18. h(3 + b) β h(b) 19. f(3 + b) β f(3 β b) 20. g(3 + b) β g(3 β b) 24. (f β g)(2) 25. (g β f)(β2) Exercise B. Simplify the following compositions. 26. (g β g)(β1) 27. (h β g)(3) 28. (f β h)(1) 29. (f β f)(1) 30. (f β g)(x) 31. (g β f)(x) 32. (f β f)(x) 33. (h β f)(x) 34. (f β h)(x) 21. g(x) + 3f(x) 22. 2g(x) + [f(x)] 2 23. g(x) / h(x)