A relation is a function if and only if no -coordinate is paired with more than one -coordinate. We test each of these four values of to see if this happens.

:

The points become:

Since -coordinate 1 is paired with two -coordinates, 2 and 9, the relation is not a function.

:

The points become:

Since -coordinate 3 is paired with two -coordinates, 0 and 9, the relation is not a function.

:

The points become:

Since -coordinates 3 and 5 are each paired with two different -coordinates, the relation is not a function.

:

The points become:

Since each -coordinate is paired with one and only one -coordinate, the relation is a function. is the correct choice.

Finding the domain is like finding out what possible  values you can plug in without getting an error message on your calculator. We are given the function:

To find the domain, first, know that you can't take the square root of a negative number.  It turns out that the minimum number you can take the square root of is zero.  Any number above zero, everything is good.  Therefore,  can equal 0 or above.  This is written as an inequality as:

Solve for  by subtracting 1 on both sides

So that's our domain!

The domain is the set of possible value for the variable. We can find the impossible values of by setting the denominator of the fractional function equal to zero, as this would yield an impossible equation.

Now we can solve for .

There is no real value of that will fit this equation; any real value squared will be a positive number.

The radicand is always positive, and is defined for all real values of . This makes the domain of  the set of all real numbers.

The domain is the set of x-values that make the function defined.

This function is defined everywhere except at , since division by zero is undefined.

The domain of a rational function is the set of all values of  for which the denominator is not equal to 0, so we set the denominator to 0 and solve for .

This is a quadratic function, so we factor the expression as , replacing the question marks with two numbers whose product is 9 and whose sum is . These numbers are , so

becomes 

,

or .

This means that , or .

Therefore, 3 is the only number excluded from the domain.

The domain of a rational function is the set of all values of  for which the denominator is not equal to 0 (the value of the numerator is irrelevant), so we set the denominator to 0 and solve for  to find the excluded values.

This is a quadratic function, so we factor the expression as , replacing the question marks with two numbers whose product is 8 and whose sum is . These numbers are , so

becomes

So either 

, in which case , 

or , in which case .

Therefore, 2 and 4 are the only numbers excluded from the domain.

In order for the function to be real, the value inside of the square root must be greater than or equal to zero. The domain refers to the possible values of the independent variable (x-value) that allow this to be true.

For this term to be real, must be greater than or equal to zero.

and stands for OR in Set Builder Notation

The graph will open to the left.  The contents inside the square root cannot be negative.  

Set the inside equal to zero.

Any number greater than one-third will be invalid, but any number below will be the domain of this function.

The domain is:   

At  , there is a hole in this function:

At all other values of x, both positive and negative, this function will be defined.