Defining functions of x

Question

Determine if the following equations define functions of x:

 

a. y=√x+5

 

b. x²+y²=5

Answer 1

Arturo, I strongly disagree with your answer for "a".  The function involves THE PRINCIPAL SQUARE ROOT (either of x or (x+5) and this is a single-valued function for every value of x in the domain.

 

This must not be confused with the situation "find the roots of y² = 4"  There, the answer is |y| = 2 which in turn gives y = ±2. (This is in conformity with the idea that an equation of degree 2 has two roots.)

Answer 2

(a)

 

To be a function, there must be only one value of y for each value of x in the domain.  Since √x can be both positive and negative, you get 2 values of y, and hence you do not have a function.

 

(b)

 

You can solve for y and get

 

y = √(5 - x²)

 

As in (a), the square root will make y multivalued, and therefore it is not a function.