Computing the inverse of a function by solving equations:

f[x_] := (x^5) + x + 1
Plot[f[x],  {x,  -5,  5}]
Plot[f[x],  {x,  -2,  2}]

Let's compute the inverse of    f    by solving the equation
f[x] = y.
For example, let's solve f[x] = 17.

f[1]
f[2]

We see that 1 is too small and 2 is too large, so we already know that the solution is in the interval [1,2].   
To place the solution more precisely, take the midpoint 1.5 and compute f(1.5), if this is too small, then the solution is in 
[1.5, 2]; 
on the other hand if f(1.5) is too large, then the solution is in 
[1, 1.5].

a  =  1. ;  b  =  2. ;
c  =  (a  +  b)/2;
If[ f[c]  ≤  17,  
        a  =  c, 
    b  =  
          c];
Print[{a, b}]

We will then continue in this manner, repeatedly dividing the interal in half and choosing either the left or right half according to the value of the function f at the midpoint.

c  =  (a  +  b)/2;
If[ f[c]  ≤  17,  
        a  =  c, 
    b  =  
          c];
Print[{a, b}]

Now we can automate the process, continuing until (b-a) is smaller than some chosen tolerance.

Clear[a, b, c]
a  =  1. ;  b  =  2. ;
While[
      b - a  >  
        10^(-10), 
c  =  (a  +  b)/
          2; 
If[ f[c]  ≤  17,  
        a  =  c, 
    b  =  
          c];
]
a

Let's check if we have found the solution to f(x) = 17.

f[a]

The number  e.

a[
        n_]  :=  (∑_(k = 0)^n) N[(1/Factorial)[k]]

For[ n =  1,  n ≤ 10, n++, 
 
      Print[a[n]]]

b[n_] := N[(((1 + (1/n)))^n)]

For[n = 1, n ≤ 5, n++, Print[b[(10^n)]]]