Calculating a Probability Distribution

Consider a vector of scores x = [x₁, x₂, ..., xₙ]. A new vector y is created by adding a constant value C to every element of x, such that yᵢ = xᵢ + C for all i. How does the output of the softmax function for y compare to the output for x?

In the context of converting a vector of raw numerical scores into a probability distribution, what is the primary role of the denominator (the summation term Σ exp(x_j)) in the softmax function softmax(x)_i = exp(x_i) / Σ exp(x_j)?

True or False: For any given non-empty vector of real numbers x, the sum of all the components of the resulting vector y = softmax(x) will always be equal to 1. The function is defined as: softmax(x)_i = exp(x_i) / Σ exp(x_j).