Answer: Yes, the relation y = x² + 12x + 31 is a function.
Explanation:
The formal definition of a function states that a function is a relation that associates the elements of a set, called the inpunt set, to the elements of another set (the ouput set) in such a way that every element of the input set is relatated with exactly one element of the output set.
The input set is represented with the independent variable and the output set is the dependent variable.
In the given relation (function), y = x² + 12x + 31 , the independent variable is x and the dependent variable is y.
You can verify that, for any x-value that you choose only one y-value (also known as image), is obtained, so that the relation does not have any ambiguity.
This is, if you, e.g., choose x = 1, y has a definite value ( 1² + 12×1 + 31 = 44) and never another different value, and the same is true for any value of x that you choose.
There is, also, a graph method to determine if a graph represents a function; it is called the vertical line test.
