hey guys. Something's been bothering me.. Well two things.. First of all, when looking at an algebraic equation and testing to see if its a function, why can X be squared, but not Y? our professor and book, gives the impression that something like: x^2 +y =1 is a function while.. -x +y^2 = 1 is NOT a function Why is this? I know that in a function, the domain can't be linked to two or more elements in the range.. is that what the second equation is saying? Is the x and y in these equations representations of the domain and range of a function? My 2nd main question is, what is the difference in saying f(x) = 1-x^2 and saying y = 1-x^2? Are they both the same thing? Why does y = f(x) anyway? I thought that f(x) was just the name of the function being presented, but then I saw my professor draw a graph, with the 'Y' letter on the yaxis replaced by the f(x) notation and was severely confused!
x^2 + y = 1 and x + y^2 = 1 ARE both functions, but not in the same variable. We call the first equation a function in y, and the second equation a function in x. Namely, for an equation to satisfy being a function in some variable, it has to be single-valued along that entire variable.
If you look at the graph of [tex] -x+y^2 = 1 [/tex] you will see that it is just a parabola resting on the y-axis, but shifted one unit down (or left, it has its vertical axis of symmetry as the x-axis). As you had mentioned in a slightly different way, a function is a rule that assigns only one value to any value of the domain (but not necessarily the other way around, a function may have several values in the domain for a value in the range). You can visually test for a function by graphing it and using the vertical line test, that is, if at any point you can draw a vertical line and intersect more than one point to the curve, then it is not a function. As with [tex] -x+y^2 = 1 [/tex] , you can draw several vertical lines and intersect the parabola twice. However, [tex] x^2+y=1 [/tex] is a function because it can be re-arranged to [tex] y = 1- x^2 [/tex] which is just a regular parabola on resting on the x-axis, but opening downwards in negative y. So they technically are both functions, but in different variables. [tex]-x+y^2 = 1 [/tex] is a function in the variable y, but not in x. Also, I want to address this : "why can X be squared, but not Y?". This is kind of a misconception. Take the equation of a circle with radius 1 centered at the origin: [tex] x^2+y^2 = 1 [/tex] , now solve for y and take the square root to get: [tex] y = ±√(1-x^2) [/tex] Each part of the circle can now be a function as they pass the vertical line test (independently). The first equation that I posted was not a function, but the second ones are functions on their own. Strictly speaking, an equation of the form f(x)=something and y=something are not the same things. A function is a set that maps some elements of the domain to some elements of the range (loosely speaking). An equation just asserts equality about two things but doesn't necessarily map in the same way.
Yes. Think of it this way. When you square x, then both x and -x get sent to the same value of y. So a horizontal line will hit the graph in two points. But that's ok, it's still a function. But if we square the y values, then two values of y will correspond to the same x. Then a vertical line will hit the graph in two places -- and therefore it can't be a function of x. But as someone pointed out, it just depends on your point of view. If you want to express x as a function of y, you just imagine the graph rotated a quarter turn. Then vertical lines become horizontal and vice versa.
Hey Newtowns Apple. The easiest way to do this graphically is to do a vertical line test in the plane of your variable that you are testing. So if its with respect to y then you check whether any line parrallel to the y-axis goes through your equation more than once. If it does its not a function in y. Apply the same thing to other variables if need be.