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## anonymous one year ago Prove that 2x-1 and e^(-x) intersect on the interval [0,1].

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1. SolomonZelman

A good calculus I approach without graphing the functions. (Be careful not to lose me, I will try my best to be clear...) ------------------- START FROM ------------------ I know $$f(x)=2x-1$$ and $$g(x)=e^{-x}$$ are both continuous functions over the interval of (−∞,+∞) and certainly over the interval of [0,1]. (NOTE: Im defining your functions as g and f for convenience) -------- ADDITIONAL PART (continuity of f and g) -------- Will you give me this continuity? If not, then here is a flashback: A polynomial function f, will be continuous over (−∞,+∞), as you know, because it is defined at every value of x. That is by definition. ((Or try to come up with a value of x in any polynomial you choose to disprove this statement. xD )) The exponential g, is also continuous everywhere. $$g(x)=e^{-x}=1/e^x$$ (and e^x will never =0 for all x) So.... Since both functions are defined for all real x, they are therefore continuous (−∞,+∞), and thus these both functions are certainly continous over [0,1]. ------------- The essence of the PROVE ------------- Since f and g are both continuous, therefore the range of each function is between it's maximum and minimum. (I assume you get why) If the ranges of f and g over the interval [0,1] intersect, $$\rm \color{red}{ and}$$ $$\bf \color{red}{ if~one~function~is~increasing~and~another~is~decreasing}$$ (but not necessarily of both decrease or both increase) then, if one increases (in this case function f with a positive slope) and the other decreases (in this case g - b/c it is an exponential decay) then you know the functions will intersect (on that interval). ---------------- HELPFUL NOTE----------------- I hope I was clear and elaborate enough to follow. Ill give you another note for the completion. Now, all you need to do is to use calculus to find the absolute maximum and absolute minimum values of f and g over the interval [0,1]. Say you find that: maximum of f is A maximum of g is C minimum of f is B minimum of g is D then the range of f is [A,B] and range of g is [C,D] if these two ranges intersect, then the functions g and f also intersect. Hope this is a helpful feedback for you. Enjoy!

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