The equation y = 3x is an algebraic equation describing the relationship between two variables, x and y. Though simple in form, the ideas and concepts behind even a simple equation such as y = 3x have evolved over the course of history.
The equation y = 3x describes how the variable y changes with another variable, x. For example, when x = 1, y = 3; when x = 100, y = 300; when x = 0, y = 0. The equation y = 3x describes the fact that for each additional x, there will be 3 more y.
Y = 3x is a linear polynomial equation, and shows direct proportionality between the two variables. Simply put this means that the equation y = 3x is represented graphically as a straight line going through the graph's origin (more on this shall be said later) and that any increase in x holds a constant proportion to the increase in y (in this case that constant is 3).
A practical example should clarify this. It is a fact that all grand pianos have 3 legs. Let x denote the number of grand pianos I find in a room. I might wonder how many piano legs there were in a room, given that I knew the number of pianos. Let y denote the number of piano legs in the room. If I were to tell you that I found 7 pianos in a shop one day, you could immediately deduce that there must have been 21 piano legs in that room, as there must have been 3 piano legs per piano, and there are 7 pianos, and 7 multiplied by 3 is 21. If there were not 21 piano legs in that room, you would no doubt ask for an explanation (perhaps a freak piano with 5 legs was present in the room). More generally, if told that there were x pianos in a room, you could say that there were 3x piano legs in that room. As y is what we have chosen to denote the number of piano legs, we could then say that y = 3x.
Alternatively, the equation y = 3x can also be seen as a mapping, from real numbers to real numbers. The same function could be defined to map from the set of complex numbers to complex numbers (by extending the domain of the function) but this additional complexity (excuse the pun!) will be dealt with later. The mapping of real numbers to real numbers would be the functional interpretation of the equation, expressed as:
F(x) = 3x and y=f(x)
Numbers put into the function (the input) are put through a predefined procedure (in this particular case multiplied by 3) and the output (resultant answer) is seen as the second member in the ordered pair of the input and output. For example, if the input were 3, then the output would be 9. We write this as f(3) = 9, which is true as 3x3 = 9. (It should be noted that the x in the previous line represents multiplication, and not the variable x. This is sometimes a source of confusion. The context in which the symbol x is used makes it clear as to whether the x stands for a variable or the multiplication symbol.) Similarly f(2) = 6, f(1) = 3.
Since f(x) = 3x (as we have defined the function f to do this) and y = 3x (from our initial equation) we can see that y = f(x). This represents the fact that y is some function of x. To go back to the piano example, the number of pianos in a room is the variable that determines the number of piano legs in that room; in other words the number of piano legs in a room is a function of the number of pianos present. The concepts of an independent and dependent variable can now be introduced.
In an equation, when one variable is determined by a different (in some cases, controlled) variable, the former variable is called the dependent variable, and the latter called the independent variable. Whilst in mathematical equations there is no causation, hence one thing never "depends" on another, in applied circumstances there are obviously times where one variable depends on a different one. For example, the condition of your car depends very much on its age (in functional terms we might say that the car's condition is a function of its age; functions are mappings of any set of objects to any other set of objects, not just numbers, so there can be functions mapping countries to their capitals, etc) but it would be nonsensical to suggest that the age of your car depends on its condition.
It was Rene Descartes who first discovered that an equation could be represented pictorially as a graph, by representing the constants that the variables in question could take as points on orthogonal axes; (axes at right angles to one another) then plotting every point fulfilling some relation (for example the points that fulfil y = 3x) onto the axes. We still call this coordinate system "Cartesian" after him. The equation y = 3x then looks like a straight line of gradient positive 3 that passes through the origin, the point (0,0) on the graph.
As our understanding of the number system has developed, complex numbers (numbers with both real and imaginary parts) were discovered, and equations such as y = 3x still have exactly the same meaning when x and y are complex. (An imaginary number is a multiple of the square root of minus one. A complex number is an ordered pair, one part of which represents a real (normal number) value, and the other which represents an imaginary value.) When expressed in Cartesian coordinates, for complex values the equation y = 3x will require 4 dimensions, and the shape produced, rather than being a straight line, will be 2 dimensional.
The equation y = 3x describes how the variable y changes with another variable, x. For example, when x = 1, y = 3; when x = 100, y = 300; when x = 0, y = 0. The equation y = 3x describes the fact that for each additional x, there will be 3 more y.
Y = 3x is a linear polynomial equation, and shows direct proportionality between the two variables. Simply put this means that the equation y = 3x is represented graphically as a straight line going through the graph's origin (more on this shall be said later) and that any increase in x holds a constant proportion to the increase in y (in this case that constant is 3).
A practical example should clarify this. It is a fact that all grand pianos have 3 legs. Let x denote the number of grand pianos I find in a room. I might wonder how many piano legs there were in a room, given that I knew the number of pianos. Let y denote the number of piano legs in the room. If I were to tell you that I found 7 pianos in a shop one day, you could immediately deduce that there must have been 21 piano legs in that room, as there must have been 3 piano legs per piano, and there are 7 pianos, and 7 multiplied by 3 is 21. If there were not 21 piano legs in that room, you would no doubt ask for an explanation (perhaps a freak piano with 5 legs was present in the room). More generally, if told that there were x pianos in a room, you could say that there were 3x piano legs in that room. As y is what we have chosen to denote the number of piano legs, we could then say that y = 3x.
Alternatively, the equation y = 3x can also be seen as a mapping, from real numbers to real numbers. The same function could be defined to map from the set of complex numbers to complex numbers (by extending the domain of the function) but this additional complexity (excuse the pun!) will be dealt with later. The mapping of real numbers to real numbers would be the functional interpretation of the equation, expressed as:
F(x) = 3x and y=f(x)
Numbers put into the function (the input) are put through a predefined procedure (in this particular case multiplied by 3) and the output (resultant answer) is seen as the second member in the ordered pair of the input and output. For example, if the input were 3, then the output would be 9. We write this as f(3) = 9, which is true as 3x3 = 9. (It should be noted that the x in the previous line represents multiplication, and not the variable x. This is sometimes a source of confusion. The context in which the symbol x is used makes it clear as to whether the x stands for a variable or the multiplication symbol.) Similarly f(2) = 6, f(1) = 3.
Since f(x) = 3x (as we have defined the function f to do this) and y = 3x (from our initial equation) we can see that y = f(x). This represents the fact that y is some function of x. To go back to the piano example, the number of pianos in a room is the variable that determines the number of piano legs in that room; in other words the number of piano legs in a room is a function of the number of pianos present. The concepts of an independent and dependent variable can now be introduced.
In an equation, when one variable is determined by a different (in some cases, controlled) variable, the former variable is called the dependent variable, and the latter called the independent variable. Whilst in mathematical equations there is no causation, hence one thing never "depends" on another, in applied circumstances there are obviously times where one variable depends on a different one. For example, the condition of your car depends very much on its age (in functional terms we might say that the car's condition is a function of its age; functions are mappings of any set of objects to any other set of objects, not just numbers, so there can be functions mapping countries to their capitals, etc) but it would be nonsensical to suggest that the age of your car depends on its condition.
It was Rene Descartes who first discovered that an equation could be represented pictorially as a graph, by representing the constants that the variables in question could take as points on orthogonal axes; (axes at right angles to one another) then plotting every point fulfilling some relation (for example the points that fulfil y = 3x) onto the axes. We still call this coordinate system "Cartesian" after him. The equation y = 3x then looks like a straight line of gradient positive 3 that passes through the origin, the point (0,0) on the graph.
As our understanding of the number system has developed, complex numbers (numbers with both real and imaginary parts) were discovered, and equations such as y = 3x still have exactly the same meaning when x and y are complex. (An imaginary number is a multiple of the square root of minus one. A complex number is an ordered pair, one part of which represents a real (normal number) value, and the other which represents an imaginary value.) When expressed in Cartesian coordinates, for complex values the equation y = 3x will require 4 dimensions, and the shape produced, rather than being a straight line, will be 2 dimensional.
