which equations or inequalities represent functions? x - y = 2

Question

which equations or inequalities represent functions? In each case, x and y are real numbers; x is in the domain and y is in the range. If answer is not a function, explain why not. 

x - y = 2

x + y < 20

y = 2x²     

y = x^{3 - 1} 

Answer 1

Inequalities do not represent functions, generally. For example, x + y < 20 is a half plane (shading beneath a dashed line).

 

x - y = 2 is a function (st. line)

y = 2x² is a function (parabola)

 

y = x^{3 - 1}  appears to be a wrongly typed function; perhaps y = x³ - 1 was intended; that is a cubic function.

 

Answer 2

A function has exactly one y-value for every x-value.

 

x - y = 2

-y = - x + 2

y = x - 2

 

No matter what value I put in for x, I'm going to get one y-value.

 

x + y < 20

y < 20 - x

 

An inequality is not a function. Let's say I say x = 5. My y-value is anything less than 15, not one value.

 

y = 2x²

 

Now, I can put in different values for x and get the same y, like 6 and -6, but that's OK.

 

y = x³ - 1

 

Again, any value for x is going to return only 1 y-value.

 

So, it's not asking, but what else would not be a function? Let's say we had x = y². That would be y = √x, and since, say √25 = 5 and -5, we now have two y-values for our x-value of 25.