Definition of a function = a Rule that assigns to each element, x, in a set called the Domain, EXACTLY ONE element, y , in the Range ......... [[ notice that the y value may be used more than once in this definition, but not the x value .........thus x = 3 y = 1, and x = 3 , y = 5 is not a function, but x = 1 , y = 7 and x = 6 y = 7 is a function ... ]]
Not all equations are functions...... for example the vertical line, x = 4, which is a linear equation in the x-y coordinate plane, is NOT a Function, Regardless of what has been suggested by others below, since x = 4 can have y = 0, y = 1, y = 3, etc...... notice, for the only element in the domain, x = 4, there are many y values in the Range....in fact, an infinite number of y values ...
..... see the link on the VLT.... vertical line test, below ......
[ I assume you are using x as the elements of the Domain, and y as elements of the Range, as is customary ] ...
A horizontal line, like y = 5 , is a Function..... for every value x in the domain, there is , in fact, only one y value associated with it... y = 5. It is ok for a function to use the y value [ range element ] repeatedly, as long as the x value in the domain is different...
For non linear equations, like ..x = y^2 , it is not a function, since x = 9 has two y values, y = 3 and y = - 3.... again violating the definition of a function, and the VLT ....
" So this would suggest that all equations are functions because where y=mx+b serves the purpose of defining a line given only two variables (its function),... " ........ as for this statement in your question, the two variables are not the function, a Relation is the rule that assigns to each x value [ independ. variable ] a certain y value [ dependent variable ].
It is a function , a stronger condition than just a relation, if it follows the Def. for a function mentioned above ....
Thus y = 2x + 3 is the Relation [ or Function, in this case ] that assigns to each x value in the domain, a y value that is 3 more than twice the x value... e.g., 2x + 3 ........ thus a point on your line , ( x, y ) , is really the point ( x, 2x + 3) , for all values of x in your domain ....
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try these links:
relations and functions: http://www.mathwarehouse.com/algebra/relation/math-function.php
VLT
http://
http://www.uiowa.edu/~examserv/mathmatters/tutorial_quiz/geometry/vertandhorizlinetests.html
http://en.wikipedia.org/wiki/Vertical_line_test
http://www.mathwarehouse.com/algebra/relation/vertical-line-test.php
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correction to Chris below...
a vertical line is a linear equation, and yet it is Not a function, so not all linear equations can be written as a function, or for that matter, a function of the other variables. However, if the linear equation is not vertical [ x = const. ] , then it is true that y = mx + b is a function ....... You are right, however, in that there are many functions from the reals to the reals, etc...and it may be difficult to write them in the form of an equation ...
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correction / comment to Tom.......
the vertical line test demonstrates that a vertical line, or a graph like x = y^2, is NOT a function, when x = element of the Domain [ independent variable... or in physics, t ], and y = element of the Range [ dependent variable ] ....I am sure the usual convention of domain and range is being used here by katrina t ......
admittedly, there may be other conventions to naming domains and ranges, as you have used them. [ e.g., x = 4 is a function over the domain y ] . Since the Domain is the "input" and Range is the "output" , I suppose in a NON STANDARD sense, your example could be considered a function.
However , I do not recall that we ever used any other convention for naming Domain and Range in any of my Math, Physics, and Engineering courses ,basic or advanced classes. No other meaning of domain and range was used than that used in a standard trig / Calculus course. ... :-[
Since having domain from the y axis, and range from the x axis is not the conventional use of Domain and Range, and it can be confusing to someone learning the concepts of a function .....I think we need to stick with the conventional use of Domain and Range, as used in most Math classes..... and as such, x = constant fails the VLT and so is NOT A FUNCTION.
[ Now, for a function that is 1 to 1... then the domain of the function is the range of the inverse function, range of f is domain of f-inv., but these are even more specialized functions then we need to answer katrina's question, and certainly does not clarify her understanding of a function. But yes, for 1 to 1 functions, they pass both the VLT and HLT, and domain and range "swap" meaning for the inverse .... but neither is a vertical line ]
check out one of many links given above, esp. the one from the Univ of Iowa
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Response to Chris........
" What I was saying is that given a linear equation and only the variables that occur in it, it is always possible to write one of them as a function of the others in the equation. Your example (x = 4) doesn't falsify the statement because y is not a variable in the equation. " .......................
Respectfully, I'm afraid that you are wrong about this. Your use of the term ' function' is incorrect here, and that is an important distinction ... since the question asked is referring to functions. You can always write one variable in a linear eqn. " in terms of the other " , but can't use the term ' function ' for x = constant. [[ sorry if this seems like I'm being picky, but a function and a relation = ' in terms of " are two different things ....... I'm sure I even misuse the term function from time to time when I teach a class, I usually try to say write y in terms of x ... it's natural to make this error ;-)... ]].
In the Cartesian coordinate plane, the line x = 4 is NOT a function, as it fails the vertical line test, a test agreed on by all Mathematicians, regardless of the missing y variable [ or even adding and subtracting a y, or a 2y, or a 3y, etc ] ........
This is the standard convention used by us Mathematicians.
Also, x = 4 can not be called a function, since it is not. It is a relation, a weaker connection between the terms then a function. The use of terminology is important here, as a relation covers a larger group of equations [ and inequalities ], while functions have an "extra" property to them, as given above.
So, you can always write a simplified "linear" equation as a relation .....but not all relations are functions. So y = mx + b, will be a function for y not = to zero, and a relation only, for y missing , since x = constant, and y, the dependent variable, takes on all values [ for y missing, we will get x = constant ........ setting y = 0 will find the equation for the vertical line, but it is misleading, since y has a range of any real # ... ]
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" After thinking about it though I can still come up with an exception to the refined version: "x = 4 + y - y" is a linear equation with two variables, but y cannot be written as a function of x. " ...................... x = 4 + y - y can be simplified to x = 4, so a Mathematician does not consider this a valid example, ....... so then x = 4 is not a function, but x = 4 + y - y is one ? Talk about confusion... there goes our VLT !!! But they are really both the same !!!! Again, at best, x = 4 + y - y is a relation, which has yet to be simplified, and not a function .........
While it is true that you would could not solve to isolate y in x = 4 + y - y ... eg, y = x - 4 + y, if you do not " collect like terms", but no one would do this in Mathematics, they would cancel out the y's to get x = 4 .
Besides, this would still not make this example a function.......... would you not simplify a problem like y = x^2 + x - 5 - x^2 - 2 + x into y = 2x - 7 before solving or graphing ? You can graph y = x^2 + x - 5 and then y = - x^2 - 2 + x then add the y coordinates at each x value to get the final y coord., but wouldn't y = 2x - 7 be easier to get that final y coordiante for each x ???
In a Math class, if you were asked to solve the above problem for x, would you say it can't be done, or give the result x = ( y + 7 ) / 2 ? I'm sure you would simplify first, and so would a Mathematician [ that's why we push students to 'collect like terms' when possible / appropriate in Basic Math thru Calculus and beyond ].
So x = 4 + y - y is not really a valid example, you would " collect like terms" first !
Hope this clears thing up Chris ...:-)