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Functions are procedures that assign a unique output to each (valid) input. Notation and Algebra of Functions
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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 – 2 x + 3 = y . Notation and Algebra of Functions
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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 – 2 x + 3 = y . Notation and Algebra of Functions name of the function
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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 – 2 x + 3 = y . Notation and Algebra of Functions name of the function the input box
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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 – 2 x + 3 = y . Notation and Algebra of Functions name of the function the input box the defining formula
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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 – 2 x + 3 = y . Notation and Algebra of Functions name of the function the input box the defining formula the output
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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 – 2 x + 3 = y . Notation and Algebra of Functions The input box “ ( ) ” holds the input to be evaluated by the formula.
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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 – 2 x + 3 = y . Notation and Algebra of Functions The input box “ ( ) ” holds the input to be evaluated by the formula. Hence f (2) means to replace x by the input (2) in the formula,
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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 – 2 x + 3 = y . Notation and Algebra of Functions The input box “ ( ) ” 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 – 2 (2) + 3
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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 – 2 x + 3 = y . Notation and Algebra of Functions The input box “ ( ) ” 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 – 2 (2) + 3 = 3 = y.
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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 – 2 x + 3 = y . Notation and Algebra of Functions The input may be other mathematics expressions. The input box “ ( ) ” 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 – 2 (2) + 3 = 3 = y.
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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 – 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 “ ( ) ” 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 – 2 (2) + 3 = 3 = y.
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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 – 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 “ ( ) ” 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 – 2 (2) + 3 = 3 = y. (The Square–Formula) (a ± b) 2 = a 2 ± 2ab + b 2 We write down the Square–Formula as a reminder below.
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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 – 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 “ ( ) ” 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 – 2 (2) + 3 = 3 = y. (The Square–Formula) (a ± b) 2 = a 2 ± 2ab + b 2 We write down the Square–Formula as a reminder below. Example A. Given f(x) = x 2 – 2x + 3, simplify the following. a. f (2a) = b. f (a + b) =
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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 – 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 “ ( ) ” 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 – 2 (2) + 3 = 3 = y. (The Square–Formula) (a ± b) 2 = a 2 ± 2ab + b 2 We write down the Square–Formula as a reminder below. Example A. Given f(x) = x 2 – 2x + 3, simplify the following. a. f (2a) = (2a) 2 – 2 (2a) + 3 b. f (a + b) =
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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 – 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 “ ( ) ” 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 – 2 (2) + 3 = 3 = y. (The Square–Formula) (a ± b) 2 = a 2 ± 2ab + b 2 We write down the Square–Formula as a reminder below. Example A. Given f(x) = x 2 – 2x + 3, simplify the following. a. f (2a) = (2a) 2 – 2 (2a) + 3 = 4a 2 – 4a + 3 b. f (a + b) =
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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 – 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 “ ( ) ” 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 – 2 (2) + 3 = 3 = y. (The Square–Formula) (a ± b) 2 = a 2 ± 2ab + b 2 We write down the Square–Formula as a reminder below. Example A. Given f(x) = x 2 – 2x + 3, simplify the following. a. f (2a) = (2a) 2 – 2 (2a) + 3 = 4a 2 – 4a + 3 b. f (a + b) = (a + b) 2 – 2 (a + b) + 3
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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 – 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 “ ( ) ” 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 – 2 (2) + 3 = 3 = y. (The Square–Formula) (a ± b) 2 = a 2 ± 2ab + b 2 We write down the Square–Formula as a reminder below. Example A. Given f(x) = x 2 – 2x + 3, simplify the following. a. f (2a) = (2a) 2 – 2 (2a) + 3 = 4a 2 – 4a + 3 b. f (a + b) = (a + b) 2 – 2 (a + b) + 3 = a 2 + 2ab + b 2 – 2a – 2b + 3
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We may form new functions by +, – , * , and / functions. Notation and Algebra of Functions
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We may form new functions by +, – , * , and / functions. Notation and Algebra of Functions Example B. Let f(x) = 3x – 4, g(x) = x 2 – 2x – 3, find f + g, f – g, f * g, and f/g.
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We may form new functions by +, – , * , and / functions. Notation and Algebra of Functions Example B. Let f(x) = 3x – 4, g(x) = x 2 – 2x – 3, find f + g, f – g, f * g, and f/g. f + g = 3x – 4 + x 2 – 2x – 3
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We may form new functions by +, – , * , and / functions. Notation and Algebra of Functions Example B. Let f(x) = 3x – 4, g(x) = x 2 – 2x – 3, find f + g, f – g, f * g, and f/g. f + g = 3x – 4 + x 2 – 2x – 3 = x 2 + x – 7
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We may form new functions by +, – , * , and / functions. Notation and Algebra of Functions Example B. Let f(x) = 3x – 4, g(x) = x 2 – 2x – 3, find f + g, f – g, f * g, and f/g. f + g = 3x – 4 + x 2 – 2x – 3 = x 2 + x – 7 f – g = 3x – 4 – (x 2 – 2x – 3)
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We may form new functions by +, – , * , and / functions. Notation and Algebra of Functions Example B. Let f(x) = 3x – 4, g(x) = x 2 – 2x – 3, find f + g, f – g, f * g, and f/g. f + g = 3x – 4 + x 2 – 2x – 3 = x 2 + x – 7 f – g = 3x – 4 – (x 2 – 2x – 3) = 3x – 4 – x 2 + 2x + 3 = –x 2 + 5x – 1
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We may form new functions by +, – , * , and / functions. Notation and Algebra of Functions Example B. Let f(x) = 3x – 4, g(x) = x 2 – 2x – 3, find f + g, f – g, f * g, and f/g. f + g = 3x – 4 + x 2 – 2x – 3 = x 2 + x – 7 f – g = 3x – 4 – (x 2 – 2x – 3) = 3x – 4 – x 2 + 2x + 3 = –x 2 + 5x – 1 f * g = (3x – 4) * (x 2 – 2x – 3)
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We may form new functions by +, – , * , and / functions. Notation and Algebra of Functions Example B. Let f(x) = 3x – 4, g(x) = x 2 – 2x – 3, find f + g, f – g, f * g, and f/g. f + g = 3x – 4 + x 2 – 2x – 3 = x 2 + x – 7 f – g = 3x – 4 – (x 2 – 2x – 3) = 3x – 4 – x 2 + 2x + 3 = –x 2 + 5x – 1 f * g = (3x – 4) * (x 2 – 2x – 3) = 3x 3 – 10x 2 – x + 12
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We may form new functions by +, – , * , and / functions. Notation and Algebra of Functions Example B. Let f(x) = 3x – 4, g(x) = x 2 – 2x – 3, find f + g, f – g, f * g, and f/g. f + g = 3x – 4 + x 2 – 2x – 3 = x 2 + x – 7 f – g = 3x – 4 – (x 2 – 2x – 3) = 3x – 4 – x 2 + 2x + 3 = –x 2 + 5x – 1 f * g = (3x – 4) * (x 2 – 2x – 3) = 3x 3 – 10x 2 – x + 12 f/g = (3x – 4)/(x 2 – 2x – 3)
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We may form new functions by +, – , * , and / functions. Notation and Algebra of Functions Example B. Let f(x) = 3x – 4, g(x) = x 2 – 2x – 3, find f + g, f – g, f * g, and f/g. f + g = 3x – 4 + x 2 – 2x – 3 = x 2 + x – 7 f – g = 3x – 4 – (x 2 – 2x – 3) = 3x – 4 – x 2 + 2x + 3 = –x 2 + 5x – 1 f * g = (3x – 4) * (x 2 – 2x – 3) = 3x 3 – 10x 2 – x + 12 f/g = (3x – 4)/(x 2 – 2x – 3) Algebraic expressions may be formed with functions notations.
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We may form new functions by +, – , * , and / functions. Notation and Algebra of Functions Example B. Let f(x) = 3x – 4, g(x) = x 2 – 2x – 3, find f + g, f – g, f * g, and f/g. f + g = 3x – 4 + x 2 – 2x – 3 = x 2 + x – 7 f – g = 3x – 4 – (x 2 – 2x – 3) = 3x – 4 – x 2 + 2x + 3 = –x 2 + 5x – 1 f * g = (3x – 4) * (x 2 – 2x – 3) = 3x 3 – 10x 2 – x + 12 f/g = (3x – 4)/(x 2 – 2x – 3) Algebraic expressions may be formed with functions notations. Example C. Let f(x) = x 2 – 2x – 3, g(x) = 2x + 1. a. Simplify f(2) – g(–3)
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We may form new functions by +, – , * , and / functions. Notation and Algebra of Functions Example B. Let f(x) = 3x – 4, g(x) = x 2 – 2x – 3, find f + g, f – g, f * g, and f/g. f + g = 3x – 4 + x 2 – 2x – 3 = x 2 + x – 7 f – g = 3x – 4 – (x 2 – 2x – 3) = 3x – 4 – x 2 + 2x + 3 = –x 2 + 5x – 1 f * g = (3x – 4) * (x 2 – 2x – 3) = 3x 3 – 10x 2 – x + 12 f/g = (3x – 4)/(x 2 – 2x – 3) Algebraic expressions may be formed with functions notations. Example C. Let f(x) = x 2 – 2x – 3, g(x) = 2x + 1. a. Simplify f(2) – g(–3) We find f(2) = (2) 2 – 2 * (2) – 3
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We may form new functions by +, – , * , and / functions. Notation and Algebra of Functions Example B. Let f(x) = 3x – 4, g(x) = x 2 – 2x – 3, find f + g, f – g, f * g, and f/g. f + g = 3x – 4 + x 2 – 2x – 3 = x 2 + x – 7 f – g = 3x – 4 – (x 2 – 2x – 3) = 3x – 4 – x 2 + 2x + 3 = –x 2 + 5x – 1 f * g = (3x – 4) * (x 2 – 2x – 3) = 3x 3 – 10x 2 – x + 12 f/g = (3x – 4)/(x 2 – 2x – 3) Algebraic expressions may be formed with functions notations. Example C. Let f(x) = x 2 – 2x – 3, g(x) = 2x + 1. a. Simplify f(2) – g(–3) We find f(2) = (2) 2 – 2 * (2) – 3 = 4 – 4 – 3
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We may form new functions by +, – , * , and / functions. Notation and Algebra of Functions Example B. Let f(x) = 3x – 4, g(x) = x 2 – 2x – 3, find f + g, f – g, f * g, and f/g. f + g = 3x – 4 + x 2 – 2x – 3 = x 2 + x – 7 f – g = 3x – 4 – (x 2 – 2x – 3) = 3x – 4 – x 2 + 2x + 3 = –x 2 + 5x – 1 f * g = (3x – 4) * (x 2 – 2x – 3) = 3x 3 – 10x 2 – x + 12 f/g = (3x – 4)/(x 2 – 2x – 3) Algebraic expressions may be formed with functions notations. Example C. Let f(x) = x 2 – 2x – 3, g(x) = 2x + 1. a. Simplify f(2) – g(–3) We find f(2) = (2) 2 – 2 * (2) – 3 = 4 – 4 – 3 = – 3
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We may form new functions by +, – , * , and / functions. Notation and Algebra of Functions Example B. Let f(x) = 3x – 4, g(x) = x 2 – 2x – 3, find f + g, f – g, f * g, and f/g. f + g = 3x – 4 + x 2 – 2x – 3 = x 2 + x – 7 f – g = 3x – 4 – (x 2 – 2x – 3) = 3x – 4 – x 2 + 2x + 3 = –x 2 + 5x – 1 f * g = (3x – 4) * (x 2 – 2x – 3) = 3x 3 – 10x 2 – x + 12 f/g = (3x – 4)/(x 2 – 2x – 3) Algebraic expressions may be formed with functions notations. Example C. Let f(x) = x 2 – 2x – 3, g(x) = 2x + 1. a. Simplify f(2) – g(–3) We find f(2) = (2) 2 – 2 * (2) – 3 = 4 – 4 – 3 = – 3 g(–3) = 2(–3) +1
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We may form new functions by +, – , * , and / functions. Notation and Algebra of Functions Example B. Let f(x) = 3x – 4, g(x) = x 2 – 2x – 3, find f + g, f – g, f * g, and f/g. f + g = 3x – 4 + x 2 – 2x – 3 = x 2 + x – 7 f – g = 3x – 4 – (x 2 – 2x – 3) = 3x – 4 – x 2 + 2x + 3 = –x 2 + 5x – 1 f * g = (3x – 4) * (x 2 – 2x – 3) = 3x 3 – 10x 2 – x + 12 f/g = (3x – 4)/(x 2 – 2x – 3) Algebraic expressions may be formed with functions notations. Example C. Let f(x) = x 2 – 2x – 3, g(x) = 2x + 1. a. Simplify f(2) – g(–3) We find f(2) = (2) 2 – 2 * (2) – 3 = 4 – 4 – 3 = – 3 g(–3) = 2(–3) +1 = –6 + 1 = –5
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We may form new functions by +, – , * , and / functions. Notation and Algebra of Functions Example B. Let f(x) = 3x – 4, g(x) = x 2 – 2x – 3, find f + g, f – g, f * g, and f/g. f + g = 3x – 4 + x 2 – 2x – 3 = x 2 + x – 7 f – g = 3x – 4 – (x 2 – 2x – 3) = 3x – 4 – x 2 + 2x + 3 = –x 2 + 5x – 1 f * g = (3x – 4) * (x 2 – 2x – 3) = 3x 3 – 10x 2 – x + 12 f/g = (3x – 4)/(x 2 – 2x – 3) Algebraic expressions may be formed with functions notations. Example C. Let f(x) = x 2 – 2x – 3, g(x) = 2x + 1. a. Simplify f(2) – g(–3) We find f(2) = (2) 2 – 2 * (2) – 3 = 4 – 4 – 3 = – 3 Hence f(2) – g(-3) = – 3 – (–5) g(–3) = 2(–3) +1 = –6 + 1 = –5
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We may form new functions by +, – , * , and / functions. Notation and Algebra of Functions Example B. Let f(x) = 3x – 4, g(x) = x 2 – 2x – 3, find f + g, f – g, f * g, and f/g. f + g = 3x – 4 + x 2 – 2x – 3 = x 2 + x – 7 f – g = 3x – 4 – (x 2 – 2x – 3) = 3x – 4 – x 2 + 2x + 3 = –x 2 + 5x – 1 f * g = (3x – 4) * (x 2 – 2x – 3) = 3x 3 – 10x 2 – x + 12 f/g = (3x – 4)/(x 2 – 2x – 3) Algebraic expressions may be formed with functions notations. Example C. Let f(x) = x 2 – 2x – 3, g(x) = 2x + 1. a. Simplify f(2) – g(–3) We find f(2) = (2) 2 – 2 * (2) – 3 = 4 – 4 – 3 = – 3 Hence f(2) – g(-3) = – 3 – (–5) = –3 + 5 = 2 g(–3) = 2(–3) +1 = –6 + 1 = –5
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b. Simplify f(x + h ) – f(x) Notation and Algebra of Functions
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b. Simplify f(x + h ) – f(x) First, we calculate f(x + h ) Notation and Algebra of Functions
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b. Simplify f(x + h ) – f(x) First, we calculate f (x + h ) = (x+h) 2 – 2 (x+h) – 3 Notation and Algebra of Functions
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b. Simplify f(x + h ) – f(x) First, we calculate f(x + h ) = (x+h) 2 – 2(x+h) – 3 = x 2 + 2xh + h 2 – 2x – 2h – 3 Notation and Algebra of Functions
42.
b. Simplify f(x + h ) – f(x) First, we calculate f(x + h ) = (x+h) 2 – 2(x+h) – 3 = x 2 + 2xh + h 2 – 2x – 2h – 3 Hence f(x+h) – f(x) = x 2 + 2xh + h 2 – 2x – 2h – 3 – (x 2 – 2x – 3) Notation and Algebra of Functions
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b. Simplify f(x + h ) – f(x) First, we calculate f(x + h ) = (x+h) 2 – 2(x+h) – 3 = x 2 + 2xh + h 2 – 2x – 2h – 3 Hence f(x+h) – f(x) = x 2 + 2xh + h 2 – 2x – 2h – 3 – (x 2 – 2x – 3) = x 2 + 2xh + h 2 – 2x – 2h – 3 – x 2 + 2x + 3 Notation and Algebra of Functions
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b. Simplify f(x + h ) – f(x) First, we calculate f(x + h ) = (x+h) 2 – 2(x+h) – 3 = x 2 + 2xh + h 2 – 2x – 2h – 3 Hence f(x+h) – f(x) = x 2 + 2xh + h 2 – 2x – 2h – 3 – (x 2 – 2x – 3) = x 2 + 2xh + h 2 – 2x – 2h – 3 – x 2 + 2x + 3 = 2xh + h 2 – 2h Notation and Algebra of Functions
45.
b. Simplify f(x + h ) – f(x) First, we calculate f(x + h ) = (x+h) 2 – 2(x+h) – 3 = x 2 + 2xh + h 2 – 2x – 2h – 3 Hence f(x+h) – f(x) = x 2 + 2xh + h 2 – 2x – 2h – 3 – (x 2 – 2x – 3) = x 2 + 2xh + h 2 – 2x – 2h – 3 – x 2 + 2x + 3 = 2xh + h 2 – 2h Notation and Algebra of Functions Composition of Functions
46.
b. Simplify f(x + h ) – f(x) First, we calculate f(x + h ) = (x+h) 2 – 2(x+h) – 3 = x 2 + 2xh + h 2 – 2x – 2h – 3 Hence f(x+h) – f(x) = x 2 + 2xh + h 2 – 2x – 2h – 3 – (x 2 – 2x – 3) = x 2 + 2xh + h 2 – 2x – 2h – 3 – x 2 + 2x + 3 = 2xh + h 2 – 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 ) – f(x) First, we calculate f(x + h ) = (x+h) 2 – 2(x+h) – 3 = x 2 + 2xh + h 2 – 2x – 2h – 3 Hence f(x+h) – f(x) = x 2 + 2xh + h 2 – 2x – 2h – 3 – (x 2 – 2x – 3) = x 2 + 2xh + h 2 – 2x – 2h – 3 – x 2 + 2x + 3 = 2xh + h 2 – 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 ○ g) (x) ≡ f ( g(x) ).
48.
b. Simplify f(x + h ) – f(x) First, we calculate f(x + h ) = (x+h) 2 – 2(x+h) – 3 = x 2 + 2xh + h 2 – 2x – 2h – 3 Hence f(x+h) – f(x) = x 2 + 2xh + h 2 – 2x – 2h – 3 – (x 2 – 2x – 3) = x 2 + 2xh + h 2 – 2x – 2h – 3 – x 2 + 2x + 3 = 2xh + h 2 – 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 ○ g) (x) ≡ f ( g(x) ). (It's read as f "circled" g).
49.
b. Simplify f(x + h ) – f(x) First, we calculate f(x + h ) = (x+h) 2 – 2(x+h) – 3 = x 2 + 2xh + h 2 – 2x – 2h – 3 Hence f(x+h) – f(x) = x 2 + 2xh + h 2 – 2x – 2h – 3 – (x 2 – 2x – 3) = x 2 + 2xh + h 2 – 2x – 2h – 3 – x 2 + 2x + 3 = 2xh + h 2 – 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 ○ g) (x) ≡ 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 ) – f(x) First, we calculate f(x + h ) = (x+h) 2 – 2(x+h) – 3 = x 2 + 2xh + h 2 – 2x – 2h – 3 Hence f(x+h) – f(x) = x 2 + 2xh + h 2 – 2x – 2h – 3 – (x 2 – 2x – 3) = x 2 + 2xh + h 2 – 2x – 2h – 3 – x 2 + 2x + 3 = 2xh + h 2 – 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 ○ g) (x) ≡ f ( g(x) ). (It's read as f "circled" g). In other words, the input for f is g(x) . Similarly, we define (g ○ f) (x) ≡ g ( f(x) ).
51.
Example C. Let f(x) = 3x – 5, g(x) = –4x + 3, simplify (f ○ g)(3), (g ○ f)(3). Notation and Algebra of Functions
52.
Example C. Let f(x) = 3x – 5, g(x) = –4x + 3, simplify (f ○ g)(3), (g ○ f)(3). (f ○ g)(3) = f( g(3) ) Notation and Algebra of Functions
53.
Example C. Let f(x) = 3x – 5, g(x) = –4x + 3, simplify (f ○ g)(3), (g ○ f)(3). (f ○ g)(3) = f( g(3) ) Since g(3) = –4(3) + 3 Notation and Algebra of Functions
54.
Example C. Let f(x) = 3x – 5, g(x) = –4x + 3, simplify (f ○ g)(3), (g ○ f)(3). (f ○ g)(3) = f( g(3) ) Since g(3) = –4(3) + 3 = –9 Notation and Algebra of Functions
55.
Example C. Let f(x) = 3x – 5, g(x) = –4x + 3, simplify (f ○ g)(3), (g ○ f)(3). (f ○ g)(3) = f( g(3) ) Since g(3) = –4(3) + 3 = –9 Hence f( g(3) ) = f (–9) Notation and Algebra of Functions
56.
Example C. Let f(x) = 3x – 5, g(x) = –4x + 3, simplify (f ○ g)(3), (g ○ f)(3). (f ○ g)(3) = f( g(3) ) Since g(3) = –4(3) + 3 = –9 Hence f( g(3) ) = f (–9) = 3 (–9) – 5 = –32 Notation and Algebra of Functions
57.
Example C. Let f(x) = 3x – 5, g(x) = –4x + 3, simplify (f ○ g)(3), (g ○ f)(3). (f ○ g)(3) = f( g(3) ) Since g(3) = –4(3) + 3 = –9 Hence f( g(3) ) = f (–9) = 3 (–9) – 5 = –32 (g ○ f)(3) = g( f(3) ) Notation and Algebra of Functions
58.
Example C. Let f(x) = 3x – 5, g(x) = –4x + 3, simplify (f ○ g)(3), (g ○ f)(3). (f ○ g)(3) = f( g(3) ) Since g(3) = –4(3) + 3 = –9 Hence f( g(3) ) = f (–9) = 3 (–9) – 5 = –32 (g ○ f)(3) = g( f(3) ) Since f(3) = 3(3) – 5 Notation and Algebra of Functions
59.
Example C. Let f(x) = 3x – 5, g(x) = –4x + 3, simplify (f ○ g)(3), (g ○ f)(3). (f ○ g)(3) = f( g(3) ) Since g(3) = –4(3) + 3 = –9 Hence f( g(3) ) = f (–9) = 3 (–9) – 5 = –32 (g ○ f)(3) = g( f(3) ) Since f(3) = 3(3) – 5 = 4 Notation and Algebra of Functions
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Example C. Let f(x) = 3x – 5, g(x) = –4x + 3, simplify (f ○ g)(3), (g ○ f)(3). (f ○ g)(3) = f( g(3) ) Since g(3) = –4(3) + 3 = –9 Hence f( g(3) ) = f (–9) = 3 (–9) – 5 = –32 (g ○ f)(3) = g( f(3) ) Since f(3) = 3(3) – 5 = 4 Hence g( f(3) ) =g (4) Notation and Algebra of Functions
61.
Example C. Let f(x) = 3x – 5, g(x) = –4x + 3, simplify (f ○ g)(3), (g ○ f)(3). (f ○ g)(3) = f( g(3) ) Since g(3) = –4(3) + 3 = –9 Hence f( g(3) ) = f (–9) = 3 (–9) – 5 = –32 (g ○ f)(3) = g( f(3) ) Since f(3) = 3(3) – 5 = 4 Hence g( f(3) ) =g (4) = –4(4) + 3 Notation and Algebra of Functions
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