Inverse Functions

Discussion in 'Algebra' started by nycmathguy, Aug 10, 2021.

  1. nycmathguy

    nycmathguy

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    A function f has an inverse
    function if and only if f is one-to-one.

    Based on the above statements, do the following 5 functions have an inverse?

    A. f(x) = x^2

    B. f(x) = x^3

    C. f(x) = | x |

    D. f(x) = sqrt{x}

    E. f(x) = 1/x
     
    nycmathguy, Aug 10, 2021
    #1
  2. nycmathguy

    MathLover1

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    A function f has an inverse function if and only if f is one-to-one.

    A. f(x) = x^2

    to find inverse, first note that f(x) =y
    y) = x^2 ...............swap variables
    x= y^2
    y= [​IMG]=>one-to-one
    upload_2021-8-11_16-29-18.png

    B. f(x) = x^3
    y= x^3
    x= y^3
    y=[​IMG]=>one-to-one
    upload_2021-8-11_16-28-36.png

    C. f(x) = | x |

    There is no inverse => not one-to-one

    D.
    f(x) = sqrt(x ) inverse is
    y= sqrt(x )
    x= sqrt(y )
    y=x^2=> not one-to-one

    E. f(x) = 1/x
    inverse =>
    y = 1/x...............swap variables
    x = 1/y...............solve for y
    y= 1/x=> is one-to-one
     
    MathLover1, Aug 11, 2021
    #2
    nycmathguy likes this.
  3. nycmathguy

    nycmathguy

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    Another easy reply.
     
    nycmathguy, Aug 11, 2021
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