kminery62

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Questions 14
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 #1
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First, I need to find f –1(x), g –1(x), and ( f o g)–1(x):

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Inverting  f (x):

Inverting g(x):

Finding the composed function:   Copyright © Elizabeth Stapel 2002-2011 All Rights Reserved

Inverting the composed function:

Now I'll compose the inverses of  f(x) and g(x) to find the formula for (g–1 o f –1)(x):

Note that the inverse of the composition (( f o g)–1(x)) gives the same result as does the composition of the inverses ((g–1 o f –1)(x)). So I would conclude that

f (x) = 2x – 1 
    y = 2x – 1 
    y + 1 = 2x 
    (y + 1)/2 = x 
    (x + 1)/2 = y 
    (x + 1)/2 =  f –1(x)

g(x) = (1/2)x + 4 
    y = (1/2)x + 4 
    y – 4 = (1/2)x 
    2(y – 4) = x 
    2y – 8 = x 
    2x – 8 = y 
    2x – 8 = g –1(x) 

( f o g)(x) = f (g(x)) = f ((1/2)x + 4) 
    = 2((1/2)x + 4) – 1 
    = x + 8 – 1 
    = x + 7

( f o g)(x) = x + 7 
    y = x + 7 
    y – 7 = x 
    x – 7 = y 
    x – 7 = ( f o g)–1(x)

(g–1 o f –1)(x) = g–1( f –1(x)) 
    = g–1( (x + 1)/2 ) 
    = 2( (x + 1)/2 ) – 8 
    = (x + 1) – 8 
    = x – 7 = (g–1 o f –1)(x)

( f o g)–1(x) = (g–1 o f –1)(x)

While it is beyond the scope of this lesson to prove the above equality, I can tell you that this equality is indeed always true, assuming that the inverses and compositions exist — that is, assuming there aren't any problems with the domains and ranges and such.

2 mai 2019