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Combining Functions Sum, Difference, Product, and Quotient Functions

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10/9/2013 Combining Functions 2 2 The important thing is not to stop questioning Albert Einstein

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10/9/2013 Combining Functions 3 3 Domain f Domain g Combining Functions f(x) g(x) + f(x)+g(x) f g (f+g)(x) = Range f = f+g Range of f(x) + g(x) and (f+g)(x) h x Range g

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10/9/2013 Combining Functions 4 How can we find the domain of f+g ? 4 Combining Functions y f(y) ? f g Ranges of f and g g(y) does not exist !! WHY ? So f(y) + g(y) does not exist !!... and neither does (f+g)(y)... since (f+g)(y) = f(y) + g(y) x Domain of f Domain of g

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10/9/2013 Combining Functions 5 How can we find the domain of f+g ? 5 Combining Functions y f(y) ? f g Ranges of f and g z g(z) g f ? x Domain of f Domain of g f(z) does not exist !! WHY ? So f(z) + g(z) does not exist !!... and neither does (f+g)(z)... since (f+g)(z) = f(z) + g(z)

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10/9/2013 Combining Functions 6 How can we find the domain of f+g ? 6 Combining Functions Ranges of f and g z g(z) g f ? x (f+g)(x) f+g Domain of f Domain of g (f+g)(x) does exist !! WHY ? Both f(x) and g(x) do exist !!... so f(x) + g(x) does exist... so (f+g)(x) = f(x) + g(x) y

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10/9/2013 Combining Functions 7 How can we find the domain of f+g ? 7 Combining Functions y Ranges of f and g z x (f+g)(x) f+g Domain of f Domain of g (f+g)(x) exists only if x lies in the domain of both f and g Dom f+g = Dom f ∩ Dom g

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10/9/2013 Combining Functions 8 8 Given functions f(x) and g(x) Define functions (f+g)(x) and (f–g)(x) for all x in the domains of BOTH f and g as (f+g)(x) = f(x) + g(x) (f–g)(x) = f(x) – g(x) Dom f+g = Dom f–g = Dom f ∩ Dom g Remember:

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10/9/2013 Combining Functions 9 9 Given functions f(x) and g(x) Define function (fg)(x) for all x in the domain of BOTH f and g as (fg)(x) = f(x) g(x) = f(x)g(x) Dom fg = Dom f ∩ Dom g Remember:

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10/9/2013 Combining Functions 10 Combining Functions Given functions f(x) and g(x) Define function (f/g)(x) for all x in the domain of BOTH f and g, with g(x) ≠ 0, as (f/g)(x) = f(x)/g(x) Dom f/g = Dom f ∩ Dom g Remember: … provided g(x) ≠ 0

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10/9/2013 Combining Functions 11 Combining Functions Examples 1. f(x) = 1 – x and g(x) = x = { x | 0 ≤ x ≤ 1 } Domain f(x) = ?Domain g(x) = ? { x | x ≤ 1 } { x | 0 ≤ x } Question: Can addition make the domain bigger ? (f+g)(x) = f(x) + g(x) 1 – x x + = Dom (f+g) = { x | x ≤ 1 } ∩ { x | 0 ≤ x } = [ 0, 1 ]

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10/9/2013 Combining Functions 12 Combining Functions Examples 2. f(x) = x + x and Domain f(x) = ?Domain g(x) = ? { x | 0 ≤ x } Question: Can addition make the domain bigger ? (f+g)(x) = f(x) + g(x) Dom (f+g) = { x | 0 ≤ x } ∩ { x | 0 ≤ x } x – x g(x) = = 2x = [ 0, ) ∞ ≠ R Domain ?

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10/9/2013 Combining Functions 13 Examples 3. Combining Functions and Smaller ? Can the domain get bigger ? = Dom f ∩ Dom g = { x | 0 ≤ x ≤ 1 } Question: f(x) = 1 – x g(x) = x Domain f(x) = ?Domain g(x) = ? { x | x ≤ 1 } { x | 0 ≤ x } (fg)(x) = f(x) g(x) ( 1 – x x ) () = Dom (fg)(x) = [ 0, 1 ]

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10/9/2013 Combining Functions 14 Examples 4. Combining Functions and Domain g(x) Domain (f/g)(x) Smaller ? = { x | 0 < x } Can the domain get bigger ? = { x | 0 ≤ x } Question: Domain f(x) f(x) = x + x x – x g(x) = (f/g)(x) x + x x – x = = Dom f ∩ Dom g; g(x) ≠ 0

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10/9/2013 Combining Functions 15 Think about it !

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