In mathematics, a function is an expression, rule, or law that defines a relationship between one variable (the independent variable) and another variable (the dependent variable). A function’s graph is the set of all points in the plane of the form (x, f(x)). A graph is a pictorial representation or diagram that represents data or values in an organized way.

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Functions & Graphs:

Functions are mathematical building blocks for designing machines, predicting natural disasters, curing diseases, understanding world economies and for keeping airplanes in the air.

Real-life Application of Functions and Graphs

Functions

A soda, snack, or stamp machine. The user puts in money, punches a specific button, and a specific item drops into the output slot. (The function rule is the product price. The input is the money combined with the selected button. The output is the product, sometimes delivered along with coins in change, if the user has entered more money than required by the function rule.)

Graphs

Graph theory can be used to visually map out all the inter-dependent chains of events that produce a specific outcome or cause a specific problem to determine the possible root causes to the problem in order to ensure that solutions directly address the causes to the problem (as opposed to merely addressing symptoms of the root causes).
One example is in Biology. Disease outbreak: Graph Theory Applied to Disease Transmission.

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The only way to learn mathematics is to do mathematics
Paul Halmos

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Operations of functions

Functions with overlapping domains can be added, subtracted, multiplied and divided. If f(x) and g(x) are two functions, then for all x in the domain of both functions the sum, difference, product and quotient are defined as follows.

Real-life Application of Functions and Graphs

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