The domain is defined as the set of possible values for the x variable. In order to find the impossible values of x, we should:

a) Set the equation under the radical equal to zero and look for probable x values that make the expression inside the radical negative:

There is no real value for x that will fit this equation, because any real value square is a positive number i.e. cannot be a negative number.

b) Set the denominator of the fractional function equal to zero and look for probable x values:

Now we can solve the equation for x:

There is no real value for x that will fit this equation.

The radical is always positive and denominator is never equal to zero, so the f(x) is defined for all real values of x. That means the set of all real numbers is the domain of the f(x) and the correct answer is .

Alternative solution for the second part of the solution:

After figuring out that the expression under the radical is always positive (part a), we can solve the radical and therefore denominator for the least possible value (minimum value). Setting the x value equal to zero will give the minimum possible value for the denominator.

That means the denominator will always be a positive value greater than 1/2; thus it cannot be equal to zero by setting any real value for x. Therefore the set of all real numbers is the domain of the f(x).

The domain is defined as the set of all possible values of the independent variable . First,  must be greater than or equal to zero, because if , thenwill be undefined. In addition, the total expression under the radical, i.e.  must be greater than or equal to zero:

That means that the expression under the radical is always positive and therefore is defined. Hence, the domain of the function is .

The domain is defined as the set of all values of x for which the function is defined i.e. has a real result. The square root of a negative number isn't defined, so we should find the intervals where that occurs:

The square of any number is positive, so we can't eliminate any x-values yet.

If the denominator is zero, the expression will also be undefined.

Find the x-values which would make the denominator 0:

Therefore, the domain is .

This function can be thought of as a composite function. It's really a quotient of two polynomials.  We know there are no restrictions on what number we plug into a polynomial, however there is one restriction on what we can plug into a quotient. You can't divide by zero. This means the domain of our function can be any real number except the ones that make the denominator equal to zero. We can rewrite this statement as . Once we factor this expression to the form  we can easily see that it is satisfied when .

This function can be thought of as a composite function. It's really a quotient of two polynomials.  We know there are no restrictions on what number we plug into a polynomial, however there is one restriction on what we can plug into a quotient. You can't divide by zero. This means the domain of our function can be any real number except the ones that make the denominator equal to zero. We can rewrite this statement as . Once we factor this expression to the form  we can easily see that it is satisfied when .  You may have been tempted to factor the top and cancel a portion of it with the bottom, but this is not allowed and leads to the trap answer .

The domain of the function are the values of  that can be plugged into the right side of the equation. The radicand must be positive and the denominator must not equal 0 so,

Thus,

There are no restrictions on the value of  when we plug it into , but when we know that logarithms can't be evaluated for anything less than or equal to zero.  So we now focus our attention to the 'stuff' inside of the natural logarithm, .  This expression must be greater than zero in order to evaluate the function. We can represent this mathematical with the inequality, .  In order to satisfy this inequality we find  

First, determine the function 

To do this, you need to insert the value of  into the value of x in 

In order to remove the fraction in the denominator, multiply everyting by (x-3)

After factoring and reducing you get the following result for :

Next, find the domain of  by setting the denominator equal to 0 and solving for x

When  the function of  will be undefined, hence it cannot be included in the domain. Thus 

Also, when you are determining the domain of  there is a restriction on the domain of  so that it is not undefined. Hence, you must set the denominator of  equal to 0 and solve for x.

When  the function of  will be undefined, hence it cannot be included in the domain of . Thus 

Hence, the domain of  is all real numbers such that 

The values of f(x) can take all negative and positive real numbers.

The function is undefined when the denominator is 0.

i.e. .

This restricts the domain to exclude that value of x.  Hence the domain is the union of open intervals from negative infinity to one-half and one-half to infinity.

There are no values that x cannot take in this polynomial.  All polynomials are one-to-one, so there are no restrictions on the values that f can take either.  So both the domain and range are all real numbers.