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# Help with composite functions please

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A)B)

Guest Dec 30, 2017
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#1
+85757
+2

A)  g(x)  =  - x/4  + 4       f(x)  =  4x   +  1/4

So

g-1 (x)

Write y for  g(x)

y  =  -x/4 + 4

y - 4  =  -x/4

4y - 16  =  - x

16 - 4y  =  x        swap  x and y

16  -  4x  =  y   =  g-1(x)

f-1(x)

Write y for  f(x)

y  =  4x  + 1/4

y - 1/4  =  4x

[ 4y - 1 ]   / 4   =  4x

[ 4y - 1 ] / 16  =  x         swap  x and y

[ 4x - 1] / 16  =  y  =  f-1(x)

(g  o  f  )(x)  =  -  [4x + 1/4] / 4  +  4   =  -x - 1/16 + 64/16   =  63/16  - x

(g o f)-1  (x)

Write y  for (g o f)(x)

y  = 63/16 -  x

y - 63/16  = -x

[ 63 - 16y] / 16  =  x    swap  x  and y

[63 - 16x ] / 16  =  y  =  (g o f )-1 (x)

( f-1 (x)  o   g-1(x)  )   =

[ 4  [16 - 4x]  -  1   ]   /  16   =

[ 64  -   16x  -  1  ]  /   16  =

(63 / 16)  -  x

We can conclude that

(g  o  f  )(x)   =   ( f-1 (x)  o   g-1(x)  )

CPhill  Dec 30, 2017
edited by CPhill  Dec 30, 2017
#2
+85757
+2

B)

Let  f(x)  =  8 * 60  =  480

Let g(x)  =  .20x  +  45

Let h(x)  =  .35x + 20

The price  charged by the first company  =   ( g  o  f  )

So we have

.20 (480) +  45  =  \$141

The price charged by the second company =  (  h   o   f  )

So we have

.35(480) + 20  =   \$188

The first company is cheaper

CPhill  Dec 30, 2017

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