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Math Help - Hi I need help on inverse functions and other things

  1. #1
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    Hi I need help on inverse functions and other things

    Hi my name is Cindy and I am new here and I need help on my math homework.

    The first one says which of the following fuctions f(x) = sqrt 19 - x and g(x) = sqrt 25 - x^2 has an inverse. Find a rule for the inverse and explain why the other function has no inverse.

    I don't understand inverses it would be nice if you could explain to me how you get the inverse.

    Also I need help on this problem : Let f(x) = x^2, g(x) = x - 2 and h(x) = 4x, find the following:

    A) f(g(h(x)) B) g(h(f(x))
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  2. #2
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    Hello, Cindy!

    Welcome aboard!
    Here's the first one . . .


    Which of the following fuctions has an inverse?
    f(x) \:= \:\sqrt{19 - x} and g(x) \:= \:\sqrt{25 - x^2}
    Find a rule for the inverse and explain why the other function has no inverse.
    To find the inverse of a function f(x):
    . . (1) Replace f(x) with y
    . . (2) Interchange the x's and y's
    . . (3) Solve for y
    . . (4) The resulting function is f^{-1}(x)


    We have: . f(x) \:=\:\sqrt{19-x}

    (1)\;\;y \;=\;\sqrt{19-x}

    (2)\;\;x\;=\;\sqrt{19-y}

    (3) .Square both sides: . x^2 \;=\;19-y\quad\Rightarrow\quad y \;=\;19-x^2

    (4)\;\text{Therefore: }\;f^{-1}(x) \;=\;19 - x^2



    We have: . g(x) \:=\:\sqrt{25-x^2}

    (1)\;\;y \;=\;\sqrt{25-x^2}

    (2)\;\;x \;=\;\sqrt{25-y^2}

    (3)\;\text{Square both sides: }\;x^2 \:=\:25-y^2\quad\Rightarrow\quad y^2 \:=\:25-x^2

    Then: . y \;=\;\pm\sqrt{25-x^2} . . . This is not a function!

    Each value of x produces two values of y.
    . . and a function must be single-valued.

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