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Find the inverse: f(x)=3, f(x)=3x^2+4x and g(x)=4x-4

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carlos485 | Student, Undergraduate | Honors

Posted April 8, 2012 at 7:44 PM via web

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Find the inverse: f(x)=3, f(x)=3x^2+4x and g(x)=4x-4

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justaguide | College Teacher | (Level 2) Distinguished Educator

Posted April 9, 2012 at 2:33 PM (Answer #1)

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It is assumed that you want the inverse functions of f(x) = 3, f(x) = 3x^2+4x and g(x)=4x-4.

For the functions f(x) = 3 and f(x) = 3x^2 + 4x the inverse function does not exist as for any value of x the value of f(x) should be unique.

For f(x) = 3, any value of x gives the result 3. The inverse of f(x) would have 3 taking on one of an infinite number of alternatives.

For f(x) = 3x^2 + 4x, 2 two values of x give the same value of f(x). The inverse of f(x) would have two options for every number x. This is not possible.

Only the inverse of g(x) = 4x - 4 can be determined.

g(x) = 4x - 4

=> x = `(g(x) + 4)/4`

substitute x with `g^-1(x)` and `g(x)` with x

=> `g^-1(x) = (x + 4)/4`

Only the inverse of g(x) = 4x - 4 exists and `g^-1(x) = (x + 4)/4.`

Sources:

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beckden | High School Teacher | (Level 1) Educator

Posted April 12, 2012 at 7:31 PM (Answer #2)

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I missed the first function f(x) = 3.  The inverse of this function is x=3.  The only way I can think to write this in function notation is `f^(-1)(y)=3` . 

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beckden | High School Teacher | (Level 1) Educator

Posted April 9, 2012 at 4:54 AM (Answer #3)

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1) `F(x) = 3x^2 + 4x`

     `y = 3x^2+4x`           Write in y = format.

     `x = 3y^2 + 4y`          Exchange x and y

     `x/3 = y^2 + 4/3y`      Divide by 3

     `x/3 + (2/3)^2 = y^2 + 4/3y + (2/3)^2`     We complete the square`(1/2b)^2 `

     `x/3 + 4/9 = (y + 2/3)^2`      Now we can take the square root

     `y+2/3 = +-sqrt(x/3+4/9)`     Note the +- this is not a function

     `y = -2/3+-sqrt(x/3+4/9)`      Subtract 2/3 from both sides

`y = (-2+-sqrt(3x+4))/3`              Simplify

` F^-1(x) = (-2+-sqrt(3x+4))/3`

    Write in inverse fromat.

2) `G(x) = 4x-4`        

    `y = 4x-4`                Write in y format

    `x = 4y - 4`              Exchange x and y

    `x + 4 = 4y`              Solve for y

     `y = 1/4x + 1`         

`G^-1(x) = 1/4x + 1`

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