Find the intersections

[LouisKing]

Hi there, I am stuck trying to find the intersections of:
f(x)= e^x and g(x)= e/x
And
f(x)= e^x and h(x)= 1/(x+1)

I know the answers are supposed to be x=1 and x=0 respectively. But I have struggle proving it with agebra.

 

[lev888]

We need to find x, such that f(x) = g(x).

 

[pka]

trying to find the intersections of:
f(x)= e^x and g(x)= e/x
And f(x)= e^x and h(x)= 1/(x+1)
I know the answers are supposed to be x=1 and x=0 respectively. But I have struggle proving it with algebra.

We want \(\displaystyle f(x)=g(0)\) so \(\displaystyle e^x=\frac{e}{x}\\xe^x=e\\xe^x-e=0\\x=~?\).

.
.
.

 

[LouisKing]

We want \(\displaystyle f(x)=g(0)\) so \(\displaystyle e^x=\frac{e}{x}\\xe^x=e\\xe^x-e=0\\x=~?\).

.
.
.

Thx for the reply
Here is are my steps, you basicly have to solve: e^x = e/x and the correct answer will be x = 1
But I have tried to get the x to one side with no result:
x = ln(e/x)
x = ln(e) - ln(x)
x = 1 - ln(x)
x + ln(x) = 1

 

[Dr.Peterson]

This kind of (transcendental) equation can't be solved algebraically; you can't isolate x and get an exact solution.

So you either use numerical methods (such as a graphing calculator or app), or just try some numbers and realize the answer is 1. It also helps to graph the LHS and see that there can only be one solution: https://www.wolframalpha.com/input/?i=x*e^x+=+e