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ghazi
is the equation of ellipse a function?
No it is not. You can pass a vertical line through it and it will pass through more than one point. So it fails the vertical line test.
okay ...what if i write it as function of y ..i mean just consider standard equation of ellipse and separate both the variables..like y in terms of x....so can't we call it function?
\[y=b * \sqrt (1-x^2/a)\]
Circle (special type of ellipse) x^2 + y^2 = r^2 y^2 = r^2 - x^2 y = +-sqrt(r^2 - x^2) y = sqrt(r^2 - x^2) or y = -sqrt(r^2 - x^2)
The last line gives it away that it can't be a function
okay...by the way what is the accurate and precised definition of function?
A function is something that maps each input to EXACTLY ONE output
With ellipses and circles, inputs map to more than one output
well that is one one onto function
no, a one to one function is each input is mapped to one unique output
not sure what you mean
if you mean multiple inputs lead to the same output, then that's still a function
if you mean a manyt-to-one function, that is a proper function. a basic example would be x^2, where, e.g. -2 and 2 map to 4
it's a function if you only get one output per input
yes that's what i mean and @jim_thompson5910 if i pass a vertical line through a parabola still it cuts parabola at two points |dw:1345840885553:dw| isn't it a fucntion ...the equation of parabola?
that's not a function since it fails the vertical line test
that's the equation x = y^2 or something similar to that
@jim_thompson5910 it could be a function if it gives you two output by one input as @08bkrishna said
btw, the parabola has to be upright to be a function Just because it's a parabola, it doesn't automatically make it a function
well x=y^2 is a parabola....no need to be upright
it isn't upright since it's sideways
in your example x is a function of y, but not vice versa
@eliassaab can you help with this...please ..and @jim_thompson5910 i said why do i need it to be upright? for considering it parabola?
yes that terminology is correct TuringTest...but confusing because x is usually the input, which means that x=y^2 isn't a function
well jim ...i have to check this out...especially your definition of function....cuz i know there is many one into function in which one input leads to many output...
@ghazi, a many to one function is where more that one input leads to the same output. this is different to a one to many, where one input leads to multiple outputs. the former is considered an equation, while the latter isn't
The whole point of the function is to predict the output. But if you get more than one output, then which output will you choose? Ex: If I have a "function" of the price of a car (y) after some number of years (x), then which price will I choose if I get more than one outputs for a particular x value?
yea agreed...thank you...it was a big confusion....special thanks to @jim_thompson5910 , @08bkrishna
\[ \frac {x^2} {a^2}+\frac {y^2} {b^2}=1 \]
For each value of x, except x=a or x=-a, y has two values. So it is not a function.