Question

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

Answer 1

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: I`m 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.
I`ll 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!