Which of the following are functions?

Question

x^2 + y^2 = 9

y = 5 + ׀x׀

y^3=x^3

Thanks. These are study questions. Please explain your answer

Answer 1

Which of the following are functions?
x^2 + y^2 = 9
This is not a function because a value of x can produce 2 different values of y. This is the equation of a circle and is not a function.

y = 5 + ׀x׀
This is a function because it passes the vertical line check. Each value of x produces only one unique value of y.
y^3=x^3

y^3=x^3
This is a function because each value of x produces only one unique value of y.

Answer 2

x^2 + y^2 = 9: No, because one value of x can correspond to two values of y (for instance, (0, 3) and (0, -3)).

y = 5 + |x|: Yes. For every value of x there is one value of |x| and hence one value of 5 + |x|, so y is a function of x.

y^3 = x^3: Yes. For every value of x there is only one possible value of y^3, and for every value of y^3 there is only one possible value for y. In fact, this equation simplifies to y = x.

(You can probably take this as given for odd-numbered powers; to prove it, suppose that a^3 = b^3, so a^3 - b^3 = 0. Now a^3 - b^3 = (a-b) (a^2 + ab + b^2), so a = b or a^2 + ab + b^2 = 0. But a^2 + ab + b^2 = (a + b/2)^2 + 3b^2/4 = (a + b/2)^2 + (b√3 / 2)^2; it can only be 0 if both (a + b/2) and (b√3 / 2) are 0, i.e. if b = 0 and a = 0, in which case a = b anyway. So if a^3 = b^3 then a = b, in other words for a given value of y^3 there is only one possible value for y.

Note that this proof doesn't hold for complex numbers, where y^3 = x^3 is not a function; in fact it has three values for each x. The step that fails is assuming that r^2 + s^2 = 0 implies that r and s are both 0; this is true for real numbers but not complex ones.)

Answer 3

only the 2nd two equations are functions. a function must pass the vertical line test, so if u were to graph it and move a vertical line across it, that line would only intersect the graph once at any given point. the first equation is a circle with radius 3 so thats not a function. the second looks like a V and its shifted up 5 (so at x=0, y=5) and then the third, solve it for y and you get y=x which is a function.

Answer 4

The rule of thumb is, if there is only one value of y regardless of what x is, then you have a function.

The first one isn't. The other two are. To help you better undersand this, I'll let you work out the explanations.

Answer 5

the 2nd one