L05 - ) = 4. Since f ( f-1 ( x )) = x Since f-1 ( f ( x ))...

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L5 – Inverse Functions and Logarithms Def – A function f with domain A is called a one-to-one function if Horizontal Line Test – Is f ( x ) = x 2 - 1 a one-to-one function? Is g ( x ) = x 3 + 1? 1
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Def – Let f be a one-to-one function with domain A and range B . Then its inverse function f - 1 has domain range and for any y in B f - 1 ( y ) = x so Note: 2
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Inverse relationships f - 1 ( f ( x )) = f ( f - 1 ( x )) = Show that f ( x ) = x 3 + 1 and g ( x ) = ( x - 1) 1 3 are inverse functions. 3
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To find the inverse of a one-to-one function: 1. 2. 3. Graphic relationship: 4
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Given the graph of f ( x ) = x 3 + 1, sketch the graph of the inverse function. 5
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Logarithmic Functions Def – If a > 0 and a 6 = 1, then f ( x ) = a x is a one-to-one increasing or decreasing function having an inverse called the logarithmic function with base a . i.e. log a ( x ) = y 6
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Sketch the graph of f ( x ) = log a ( x ) based on the inverse function f - 1 ( x ) = a x . 7
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Properties of logarithms ( x and y positive) 1. log a ( xy ) = 2. log a ( x y ) = 3. log a ( x y
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Unformatted text preview: ) = 4. Since f ( f-1 ( x )) = x Since f-1 ( f ( x )) = x 8 Evaluate: 1. log 3 ( 1 9 2 ) 2. 5 log 5 (4)-log 5 (3) Write as a single logarithm: log 4 ( x-1)-1 2 log 4 ( x + 3) Solve for x : 2log( x ) = 6 log 2 ( x + 6) = 2log 2 ( x ) 9 The common logarithm has base 10 and is denoted The natural logarithm has base e and is denoted Properties ln x = y f ( f-1 ( x )) = x f-1 ( f ( x )) = x 10 Solve for x : 1. ln(2 x-4) = 3 2. e x 2 = 5 3. 2ln( x )-ln(3-x ) = ln( 1 2 ) + ln(8) 11 Sketch the graphs: y = ln( x ) y = ln( x + 2)-1 12 Find the inverse of y = ln( x + 2)-1. 13 Change of base formula For any a > 0 and a 6 = 1 log a ( x ) = ln( x ) ln( a ) 14 Where do the graphs intersect? y = log 2 ( x + 2); y = log 4 (8 x ) 15...
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L05 - ) = 4. Since f ( f-1 ( x )) = x Since f-1 ( f ( x ))...

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