let A=R-{3} , B=R- {1}. If f : AB be defined by f(x)= x-2/x-3, & all x EA. Show that f is bijective

Question

let A=R-{3} , B=R- {1}.
If f : AB be defined by
f(x)= x-2/x-3, & all x EA.
Show that f is bijective

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Lyla 6 days 2021-09-10T15:33:26+00:00 1 Answer 0 views 0

Answers ( )

    0
    2021-09-10T15:34:48+00:00

    Answer:

    ANSWER

    A=R−{3}

    B=R−{1}

    f:A→B

    f(x)= x−3x−2

    f(x 1)=f(x 2 )x 1 −3x 1 −2 = x 2

    −3

    x

    2

    −2

    (x

    2

    −3)(x

    1

    −2)=(x

    2

    −2)(x

    1

    −3)

    x

    1

    x

    2

    −3x

    1

    −2x

    2

    +6=x

    1

    x

    2

    −3x

    2

    −2x

    1

    +6

    −3x

    1

    −2x

    2

    =−3x

    2

    −2x

    1

    −x

    1

    =−x

    2

    x

    1

    =x

    2

    So, f(x) is one-one

    f(x)=

    x−3

    x−2

    y=

    x−3

    x−2

    y(x−3)=x−2

    yx−3y=x−2

    yx−x=3y−2

    x(y−1)=3y−2

    x=

    (y−1)

    3y−2

    f(x)=

    x−3

    x−2

    =

    y−1

    3y−2

    −3

    y−1

    3y−2

    −2

    =

    y−1

    3y−2−3(y−1)

    y−1

    3y−2−2(y−1)

    =

    3y−2−3y+3

    3y−2−2y+2

    =

    −2+3

    3y−2y

    =y

    f(x)=y

    f(x) is onto.

    So f(x) is bijective and invertible

    f(x)=

    x−3

    x−2

    y=

    x−3

    x−2

    x=

    y−3

    y−2

    x(y−3)=y−2

    xy−3x=y−2

    xy−y=3x−2

    y(x−1)=3x−2

    y=

    x−1

    3x−2

    f

    −1

    (x)=

    x−1

    3x−2

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