We can see that it is a function because it passes the vertical line test. We can also see
that it assigns only one x value to each y value. Thus, it is a one-to-one function.
Again from precalculus, we can see graphically whether a function is a one-to-one
function by using the horizontal line test:
Any horizontal line we draw through the graph of the function y = x^{3} passes through
only one point, so it must be assigning only one x value to each y, and can therefore
be considered a one-to-one function. Horizontal lines through y = x^{2} + 2 pass through
more than one point, so this function fails the horizontal line test.
In summary, for a rule to be a function, its graph must pass the vertical line test. To be a
one-to-one function, it must pass both the vertical line test and the horizontal line test.
Functional Notation
In this guide, we will often give functions names such as f (x), g(x), h(x), etc. For
example, when we say "f (x) = x^{2} + 2", we mean for f (x) to refer to the rule that assigns the number y = x^{2} + 2 to any real number x.
Two Types of Functions: Rational and Polynomial
As we proceed, two types of functions to be aware of are polynomial
functions and rational functions.