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Let f(x)=3x+1. Find each value. 1. (f^-1 `@` f)(5) 2. (f^-1 `@` f)(6)

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hannahstakes | Student, Grade 10 | eNotes Newbie

Posted May 15, 2013 at 6:05 AM via web

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Let f(x)=3x+1. Find each value.

1. (f^-1 `@` f)(5)

2. (f^-1 `@` f)(6)

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pramodpandey | College Teacher | Valedictorian

Posted May 15, 2013 at 6:16 AM (Answer #1)

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We have given

f(x)=3x+1

It is linear function so its inverse will exist.

`f^(-1)(x)=(x-1)/3`

`(f^(-1)o f)(x)=x`

`(f^(-1)o f)(5)=5`

`(f^(-1)o f)(6)=6`

``

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oldnick | Valedictorian

Posted May 15, 2013 at 2:06 PM (Answer #2)

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An algebraic rule, set that functional product of one to one function is again a one to one function.

Indeed:   `f: A---> B` ; `g:B--->C` so:

`AAy inB ---> EEx inA | y=f(x) `

`AA z in C ---> EE y in B | z=g(y)`  (Surriettive property)

`f(x)=f(x_0) rArr x= x_0`  `g(y)=g(y_0) rArr y=y_0` (innietive rules)

So if `z=z_0,EE y,y in A_0 | g(y) = z, g(y_0)= z_0`  (1) 

For suriretvie  property of `g(y)` 

Further  `EE x. x_0 in A | f(x)=y; f(x_0)= y_0`         (2)

 Cause inniective rules  (1) reports: `y=y_0` , and   agian by (2)

  `x=x_0`

 

Now in the function we have to find  is product of

  `f: RR ----> RR`

and

`f^(-1): RR-----> RR`

So the function  product `f@ f^(-1)` (That generally doesn't collides with identity) is product of `y=f(x)= 3x +1`  and it's inverse, `x=f^(-1)(y)`,that, being  both  linear, are one to one functions.

So that,  their product is again a one to one function, as we have shown above, that means  `x_0 in RR` is image only and only of itself.

In conclusion we get:   `f@ f^(-1) (5)=5;f@f(-1) (6)= 6`    ``    

 

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oldnick | Valedictorian

Posted May 15, 2013 at 2:20 PM (Answer #3)

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`ERRRRATTA-CCOORRRRIGE:`

 

at relation (1) :

"So if    `z=z_0,EE y,y inA_0|g(y)=z,g(y_0)=z_0`   (1) "

You should read instead:

" So if  `z=z_0,EE y,y_0 in A| g(y)=z,g(y_0)=z_0`    (1)"

 

 

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