Final answer:
A function ensures that every input has exactly one output. It can represent relationships, such as those found in equations of lines or growth rates. The Vertical Line Test can be used to confirm if a relationship is a function by checking that a graph intersects vertical lines at only one point.
Step-by-step explanation:
To determine if something is a function, you need to verify if every input (often represented by the variable x) has exactly one output (often represented by the variable y). In terms of economic models, functions represent relationships between variables such as price and demand. A mathematical function can be as simple as a definition, such as 'Professor = Adam Smith', implying a one-to-one relationship.
To illustrate further, let’s consider the equation of a line, an example of a function. It has a slope (rate of change) and an intercept (starting value). When you change the slope or the intercept of a line, you are manipulating the function. Computing and interpreting a growth rate involves understanding the percentage change, which is another function at work.
When you read or manipulate a graph, you are often observing the graphical representation of a function. The critical test for a function is that for any given x-value (input), there should be only one corresponding y-value (output). This is sometimes called the Vertical Line Test because in a graph, a vertical line at any x-value should intersect the function's curve at only one point.
From the examples given about even and odd functions, we see more properties of functions. If a function is even, its graph is symmetric about the y-axis, like x². An odd function, such as x³, has a graph that is symmetric about the origin. The product of two even functions, or two odd functions, will result in an even function. However, when an odd function is multiplied by an even function, the resulting function is odd.