If a function is either always increasing or always decreasing then it is one to one,
Do you understand what "one to one" means? A function, f, is one to one if and only if f(x)= f(y) implies x= y. If f is increasing, we can use "proof by contradiction": Suppose f(x)= f(y) with . Then either x> y so that f(x)> f(y) or x< y f(x)< f(y), either contradicting f(x)= f(y). The same argument works if f is decreasing.