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Math Help - Functions

  1. #1
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    Functions

    Let the function f : [3,1) -> R be defined by f(z) = (z −3)^2 +5, and
    g : R -> R be defined by g(x) = 7 e^x.
    (a) Find ran f.
    (b) Find g(ran f).
    (c) Let B := ran f and C := g(ran f). Write explicitly the expression for (g)^−1(h) for h E C,
    and the expression for (f)^−1(x) for x E B.
    (d) Write explicitly the expression for (g o f)^−1(h) for h E C.
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  2. #2
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    Quote Originally Posted by luckyc1423 View Post
    Let the function f : [3,1) -> R be defined by f(z) = (z −3)^2 +5, and
    g : R -> R be defined by g(x) = 7 e^x.
    (a) Find ran f.
    What is [3,1)=\{\}?
    I assume [1,3).

    The range are all y\in \mathbb{R} such that f(z)=y for some z\in [1,3).
    Thus,
    f(z)=y
    (z-3)^2+5=y
    (z-3)^2=y-5
    To have a solution we need that, y-5\geq 0.
    Thus,
    z=3\pm \sqrt{y-5}
    We need that,
    1\leq 3\pm \sqrt{y-5} <3
    -2\leq \pm \sqrt{y-5}<0
    We see that + cannot work.
    Thus,
    -2\leq -\sqrt{y-5}<0 this tells us that y\not = 5.
    It is equivalnet to squaring both sides,
    4\geq y-5<0
    9\geq y < 5
    Note it satisfies the condition we have before y-5\geq 0.
    (b) Find g(ran f).
    This is defined as,
    g: (5,9]\to \mathbb{R}
    g(x)=7e^x
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