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Math Help - Function determination from graph.

  1. #1
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    Function determination from graph.

    My understanding is that I can find out whether a function is injective, surjective / bijective by plotting the graph and noticing how many intersections there at a given point. I'm working through a few examples but I am stuck on this one.

    f(x) = 10^6 - 2^-x

    It does not seem practical to plot this curve (I'm not using a graphing calculator).

    Do I just use a particularly large scale and plot points as normal?
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  2. #2
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    Rather than graphing it would be better, in general, to think about what "surjective" and "injective" mean.

    "Surjective", also called "onto", just means that f(x) takes on every possible value- which is the same as saying we can solve the equation f(x)= 10^6- 2^{-x}= y for all y. Here, we would solve by:
    1) subtract 10^6 from both sides: -2^{-x}= y- 10^6
    2) multiply both sides by -1: 2^{-x}= 10^6- y
    3) take the logarithm of both sides: -x= log(10^6- y)
    4) multiply both sides by -1: x= -log(10^6- y)

    Now, can we do that for all y? The only possible problem is with that logarithm. log(x) is only defined for x> 0 so we must have 10^6- y> 0 which means y< 10^6.
    No, this function is NOT surjective on all real numbers. Its value is never larger than 10^6.

    "Injective", also called "one-to-one", means that two different values of x must give different values of f(x) so you need to ask "is it possible to have f(x)= f(y) without x=y?"

    Look at f(x)= 10^6- 2^{-x}= 10^6- 2^{-y}= f(y).
    The obvious first step is to subtract 10^6 from both sides from both sides: -2^{-x}= -2^{-y}
    Now multiply both sides by -1: 2^{-x}= 2^{-y}
    Take the logarithm of both sides: -xlog(2)= -ylog(2)
    Finally, divide both sides by -log(2): x= y.

    That says that if f(x)= f(y), then we must have x= y. Yes, this function is injective.
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  3. #3
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    Thanks for you answer, I appreciate it. I understand the idea behind injective, surjective and bijective. It is how one domain is mapped to its codomain (or range).

    The thing is I'm asked to solve the question by drawing the graph and then determining from this whether it's injective, surjective or bijective. Any hints on how to draw this? Would I just use a particularly large range in the scale?
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