Slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising. If you continue browsing the site, you agree to the use of cookies on this website. See our User Agreement and Privacy Policy.

Slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising. If you continue browsing the site, you agree to the use of cookies on this website. See our Privacy Policy and User Agreement for details.

Like this presentation? Why not share!

- 1.3 sign charts and inequalities by math123c 1588 views
- 4 5 fractional exponents by math123b 1141 views
- 2.4 introduction to logarithm by math123c 1267 views
- 3.4 ellipses by math123c 762 views
- 1.6 sign charts and inequalities i by math260 3539 views
- 1.6 inverse function (optional) by math123c 848 views

1,381 views

Published on

No Downloads

Total views

1,381

On SlideShare

0

From Embeds

0

Number of Embeds

208

Shares

0

Downloads

0

Comments

0

Likes

1

No embeds

No notes for slide

- 1. Notation and Algebra of Functions
- 2. Functions are procedures that assign a unique output to each (valid) input. Notation and Algebra of Functions
- 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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 SquareβFormula) (a Β± b) 2 = a 2 Β± 2ab + b 2 We write down the SquareβFormula as a reminder below.
- 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 SquareβFormula) (a Β± b) 2 = a 2 Β± 2ab + b 2 We write down the SquareβFormula 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. 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 SquareβFormula) (a Β± b) 2 = a 2 Β± 2ab + b 2 We write down the SquareβFormula 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. 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 SquareβFormula) (a Β± b) 2 = a 2 Β± 2ab + b 2 We write down the SquareβFormula 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. 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 SquareβFormula) (a Β± b) 2 = a 2 Β± 2ab + b 2 We write down the SquareβFormula 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. 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 SquareβFormula) (a Β± b) 2 = a 2 Β± 2ab + b 2 We write down the SquareβFormula 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. We may form new functions by +, β , * , and / functions. Notation and Algebra of Functions
- 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. 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. 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. 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. 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 = βx 2 + 5x β 1
- 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 = βx 2 + 5x β 1 f * g = (3x β 4) * (x 2 β 2x β 3)
- 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 = βx 2 + 5x β 1 f * g = (3x β 4) * (x 2 β 2x β 3) = 3x 3 β 10x 2 β x + 12
- 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 = βx 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. 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 = βx 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. 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 = βx 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(β3)
- 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 = βx 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(β3) We find f(2) = (2) 2 β 2 * (2) β 3
- 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 = βx 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(β3) We find f(2) = (2) 2 β 2 * (2) β 3 = 4 β 4 β 3
- 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 = βx 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(β3) We find f(2) = (2) 2 β 2 * (2) β 3 = 4 β 4 β 3 = β 3
- 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 = βx 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(β3) We find f(2) = (2) 2 β 2 * (2) β 3 = 4 β 4 β 3 = β 3 g(β3) = 2(β3) +1
- 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 = βx 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(β3) We find f(2) = (2) 2 β 2 * (2) β 3 = 4 β 4 β 3 = β 3 g(β3) = 2(β3) +1 = β6 + 1 = β5
- 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 = βx 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(β3) 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 (β5) g(β3) = 2(β3) +1 = β6 + 1 = β5
- 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 = βx 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(β3) 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 (β5) = β3 + 5 = 2 g(β3) = 2(β3) +1 = β6 + 1 = β5
- 38. b. Simplify f(x + h ) β f(x) Notation and Algebra of Functions
- 39. b. Simplify f(x + h ) β f(x) First, we calculate f(x + h ) Notation and Algebra of Functions
- 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. 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. 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. 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. 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. 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. 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. 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. 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 "circled" g).
- 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 "circled" g). In other words, the input for f is g(x) .
- 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 "circled" g). In other words, the input for f is g(x) . Similarly, we define (g β f) (x) β‘ g ( f(x) ).
- 51. Example C. Let f(x) = 3x β 5, g(x) = β4x + 3, simplify (f β g)(3), (g β f)(3). Notation and Algebra of Functions
- 52. Example C. Let f(x) = 3x β 5, g(x) = β4x + 3, simplify (f β g)(3), (g β f)(3). (f β g)(3) = f( g(3) ) Notation and Algebra of Functions
- 53. Example C. Let f(x) = 3x β 5, g(x) = β4x + 3, simplify (f β g)(3), (g β f)(3). (f β g)(3) = f( g(3) ) Since g(3) = β4(3) + 3 Notation and Algebra of Functions
- 54. Example C. Let f(x) = 3x β 5, g(x) = β4x + 3, simplify (f β g)(3), (g β f)(3). (f β g)(3) = f( g(3) ) Since g(3) = β4(3) + 3 = β9 Notation and Algebra of Functions
- 55. Example C. Let f(x) = 3x β 5, g(x) = β4x + 3, simplify (f β g)(3), (g β f)(3). (f β g)(3) = f( g(3) ) Since g(3) = β4(3) + 3 = β9 Hence f( g(3) ) = f (β9) Notation and Algebra of Functions
- 56. Example C. Let f(x) = 3x β 5, g(x) = β4x + 3, simplify (f β g)(3), (g β f)(3). (f β g)(3) = f( g(3) ) Since g(3) = β4(3) + 3 = β9 Hence f( g(3) ) = f (β9) = 3 (β9) β 5 = β32 Notation and Algebra of Functions
- 57. Example C. Let f(x) = 3x β 5, g(x) = β4x + 3, simplify (f β g)(3), (g β f)(3). (f β g)(3) = f( g(3) ) Since g(3) = β4(3) + 3 = β9 Hence f( g(3) ) = f (β9) = 3 (β9) β 5 = β32 (g β f)(3) = g( f(3) ) Notation and Algebra of Functions
- 58. Example C. Let f(x) = 3x β 5, g(x) = β4x + 3, simplify (f β g)(3), (g β f)(3). (f β g)(3) = f( g(3) ) Since g(3) = β4(3) + 3 = β9 Hence f( g(3) ) = f (β9) = 3 (β9) β 5 = β32 (g β f)(3) = g( f(3) ) Since f(3) = 3(3) β 5 Notation and Algebra of Functions
- 59. Example C. Let f(x) = 3x β 5, g(x) = β4x + 3, simplify (f β g)(3), (g β f)(3). (f β g)(3) = f( g(3) ) Since g(3) = β4(3) + 3 = β9 Hence f( g(3) ) = f (β9) = 3 (β9) β 5 = β32 (g β f)(3) = g( f(3) ) Since f(3) = 3(3) β 5 = 4 Notation and Algebra of Functions
- 60. Example C. Let f(x) = 3x β 5, g(x) = β4x + 3, simplify (f β g)(3), (g β f)(3). (f β g)(3) = f( g(3) ) Since g(3) = β4(3) + 3 = β9 Hence f( g(3) ) = f (β9) = 3 (β9) β 5 = β32 (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. Example C. Let f(x) = 3x β 5, g(x) = β4x + 3, simplify (f β g)(3), (g β f)(3). (f β g)(3) = f( g(3) ) Since g(3) = β4(3) + 3 = β9 Hence f( g(3) ) = f (β9) = 3 (β9) β 5 = β32 (g β f)(3) = g( f(3) ) Since f(3) = 3(3) β 5 = 4 Hence g( f(3) ) =g (4) = β4(4) + 3 Notation and Algebra of Functions
- 62. Example C. Let f(x) = 3x β 5, g(x) = β4x + 3, simplify (f β g)(3), (g β f)(3). (f β g)(3) = f( g(3) ) Since g(3) = β4(3) + 3 = β9 Hence f( g(3) ) = f (β9) = 3 (β9) β 5 = β32 (g β f)(3) = g( f(3) ) Since f(3) = 3(3) β 5 = 4 Hence g( f(3) ) =g (4) = β4(4) + 3 = β13 Notation and Algebra of Functions
- 63. Note: They are different! Example C. Let f(x) = 3x β 5, g(x) = β4x + 3, simplify (f β g)(3), (g β f)(3). (f β g)(3) = f( g(3) ) Since g(3) = β4(3) + 3 = β9 Hence f( g(3) ) = f (β9) = 3 (β9) β 5 = β32 (g β f)(3) = g( f(3) ) Since f(3) = 3(3) β 5 = 4 Hence g( f(3) ) =g (4) = β4(4) + 3 = β13 Notation and Algebra of Functions
- 64. f(x) = 2 β 3x g(x) = β2x 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(β1 ) 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)(β2) Exercise B. Simplify the following compositions. 26. (g β g)(β1) 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)

No public clipboards found for this slide

×
### Save the most important slides with Clipping

Clipping is a handy way to collect and organize the most important slides from a presentation. You can keep your great finds in clipboards organized around topics.

Be the first to comment