Question

how to determine whether the equation defines y as a function of x. is it a function yes or no to the problem: x=y^2-6 and problem y=x^2+6

Answer 1

$A\ function\ is\ a\ special\ relationship\ between\ values:\\\\ to\ each\ value\ of\ x\ the\ rule\ assigns\ one\ and\ only\ one\ value\ of\ y.\\------------------------------\\y=x^2+6\ \ \ it\ is\ the\ function\\\\x=0\ \ \ \Rightarrow\ \ \ y=0^2+6=6\\x=-3\ \ \ \Rightarrow\ \ \ y=(-3)^2+6=9+6=15\\x=5\ \ \ \Rightarrow\ \ \ y=5^2+6=25+6=31\\------------------------------\\x=y^2-6\ \ \ it\ is\ not\ the\ function$

$for\ example:\\x=3\ \ \ \ \Rightarrow\ \ \ \ 3=y^2-6\ \ \ \ \Rightarrow\ \ \ y^2=9\ \ \ \ \Rightarrow\ \ \ (y=3\ \ \ or\ \ \ y=-3)\\x=10\ \ \ \Rightarrow\ \ \ 10=y^2-6\ \ \ \Rightarrow\ \ \ y^2=16\ \ \ \Rightarrow\ \ \ (y=4\ \ \ or\ \ \ y=-4)\\x=75\ \ \ \Rightarrow\ \ \ 75=y^2-6\ \ \ \Rightarrow\ \ \ y^2=81\ \ \ \Rightarrow\ \ \ (y=9\ \ \ or\ \ \ y=-9)$