What is the cardinality of the set of real solutions of e^x + x = 1?

Concept:

To find the critical points of a function  we put  = 0.

Explanation:  

We are looking for real values of    that satisfy this equation. Let’s rewrite this as 

To find the real solutions, we need to investigate the points where these two functions intersect, i.e., where,

As   , , and .

So, there is no solution in this region.

For large positive   :

As   , , and   .

Again, there is no solution in this region.

Check around   :

At , , which satisfies the equation.

So,   is a solution.

We can check the behavior of the function .

The derivative of this function is 

Since for all real , the function is strictly increasing. Therefore, it can only cross   at one point,

which implies that the solution   is the only real solution.

Since there is only one real solution, the cardinality of the set of real solutions is 1.

Thus, the correct answer is option 2).