Question

Explain why the equation x^2+y^2=1 does not define y as a function of x

Answer 1

A vertical line crosses the graph of it in two places. When that is true, the relation does not meet the definition of a function. A function must be single-valued for each distinct value of the independent variable.

Answer 2

Function by definition can have only one possible y value for any given x value.

So take x = sqrt (5)
Substituting into the equation we get:
y^2 = (sqrt 5)^2 - 1
y^2 = 5 - 1
y^2 = 4

So what y value, when it is squared, gives us 4? It could be either 2 or -2. So there are two y-values that match one x value and the equation is not a function.

x^2 + y^2 = 4 is actually the equation for a circle with center at the origin and a radius of 2. Try graphing it by hand by choosing x = 0, -2, and 2 to get four points on the circle.  Unless they specify that X Does not equal to one, then this isn't a function. Say X DOES = 1 then you'd have 1/(1^2-1) which would give you 1/0 which isn't a function. So like the other have said, it IS a function UNLESS x=1 Hope that helps!