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Math Help - One-one function

  1. #1
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    One-one function

    Find the derivative of h(x) where h(x)=\frac{1}{f(x)} and f(x)=tanx+1, 0<x<\frac{\pi}{2}

    Hence show that h(x) is a one-one function

    I have found the derivative of h(x) to be -\frac{sec^2x}{(tanx+1)^2} but am confused on how i could use this to show h(x) is one-one
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  2. #2
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    If a function is either always increasing or always decreasing then it is one to one,
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  3. #3
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    oh... so u mean that the gradient is always decreasing or increasing, then it is one to one?
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  4. #4
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    Quote Originally Posted by HallsofIvy View Post
    If a function is either always increasing or always decreasing then it is one to one,
    sorry but i didnt quite get what you meant
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  5. #5
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    Do you understand what "one to one" means? A function, f, is one to one if and only if f(x)= f(y) implies x= y. If f is increasing, we can use "proof by contradiction": Suppose f(x)= f(y) with x\ne y. Then either x> y so that f(x)> f(y) or x< y f(x)< f(y), either contradicting f(x)= f(y). The same argument works if f is decreasing.
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  6. #6
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    Quote Originally Posted by Punch View Post
    oh... so u mean that the gradient is always decreasing or increasing, then it is one to one?
    No, I mean that if the function is increasing or decreasing (it gradient (derivative) is non-negative or non-positive) then it is one to one.
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  7. #7
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    Quote Originally Posted by HallsofIvy View Post
    No, I mean that if the function is increasing or decreasing (it gradient (derivative) is non-negative or non-positive) then it is one to one.
    Actually, to be technically precise, the derivative would have to be positive or negative; it could only be zero at isolated points and not on any interval.
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  8. #8
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    For a function to be one-to-one, the function must pass the vertical line test and the horizontal line test for all values of x.
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