## Functions, Graphs, & Equations – Introduction

A function is a relation from inputs to possible outputs where each input is related to exactly one output, basically it is a rule that transforms numbers to other numbers. Functions show the relationships between different quantities. A real world example of a function would be taking a particular item say bread and relate it to other objects such as a yardstick of measurement here we use money. Objects can be called sets (a collection of things) where the order of loaves does not matter. For instance, the set of the first four loaves denoted by {loaf 1, loaf 2, loaf 3, loaf 4}. We are not multiplying f(x), but define the rule f by the equation f(x) = y. Suppose f is the function which measures revenue at the bread stall by associating both variables, taking the # of bread sold as the input and gives $ made as the output. If each loaf cost $4.50 then f(x) is = 4.5x and selling 5 loaves profits f(5) = 4.5(5) = 22.5. If I sell 10 loaves then f(10) = 4.50(10) = 45 dollars. Here 10 is the input and 45 is the output. Where f is the function, x is input, and y is output. All we do is plug in the figures for x and y.

A graph represents connections or interrelations among two or more variables by coordinates and the series of points form a curve or line, each represents a value of a given function. We graph functions the same way that we graph equations. Where one quantity depends on another quantity, for instance if you sell bread your pay in $ depends on the # of hours you work. If I can sell 5 loaves in 1 hour, you find if I had 2 hours I can sell 10 loaves. The amount of loaves I can sell is proportionate to # of hours of work. Graphing functions is helpful to model a real world problem, modelling finds the relationships between quantities. We can graph quadratic, linear, & simultaneous equations, the only real difference is how many points you need to plot. A linear equation y = 5x + 1 where you just have x and not x squared to make a straight line. A basic quadratic equation is y = x^{2 }where two+ points are required to graph the line. Quadratic equations originated from the Babylonians, for the use of taxes.

An equation shows equality of two expressions, a form of symbols on either side of an equal sign, for instance 10 + 3 + 2 = 15. Step one involves solving equations that contain unknown variables and learn how to identify a solution with given numbers as well as by using inverse operations. Equations are solved with multiplication such as 5x = 25, the # you multiply with 5 to get 25 is 5, so 5 replaces x to satisfy the equation and you get 5 x 5 = 25. This works the same for subtraction such as 5x = 0, the x must be 5 to satisfy the equation 5-5 = 0. For harder to solve equations we use inverse operations which reverse an operation, an operation can be (multiplication, division), (subtraction, or addition). Those in brackets are inverses.

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