Prove the uniqueness of a solution of an equationProve that 3+1/(x+1) has an unique solution
We have to prove that 3+1/(x+1) has a unique solution.
=> [3(x +1) + 1] / (x+1)
=> [3x + 3 + 1] / (x +1)
=> [3x + 4] / (x + 1)
We see that the maximum power of x in the function is 1. Therefore it can have only a single unique solution.
In order to prove that an equation has an unique solution, we'll have to prove that the function is one to one function.
For this reason, we'll have to prove that the function is monotonous. We'll calculate the first derivative:
f'(x)=-1/(x+1)^2<0 for any x from R, so f is decreasing then it is an one to one function.
If the function is injective, any parallel line to x axis will intercept the graph of the function in a single point, so the solution of the equation is unique.