Expert: Steve Holleran - 8/13/2008

Question
QUESTION: What is the domain of:

y = the square root of x + 1 / x + 5

+

the square root of x + 5 / x + 1

ANSWER: Hi Mandi,

This one looks very intimidating, but I think we can work through it.

I use something called a "sign chart" to do these.  A sign chart is just a small number line that is broken into intervals, and then test numbers in the intervals are used in the expression to find if its postive or negative.

In your problem, we have to find the domain of each radical, and then take the most restrictive for the final domain.

For the first radical, the expression under the radical is

         x+1 / x+5

To do this sign chart, the "split points" are the values which make each part of the fraction = 0 --> x = -1 and x = -5, so mark these off:

         ----------|----------|-----------
         -5          -1

Now just pick numbers in each interval, and put them into the fraction to get the resulting sign.  For example, 0 is in the interval to the right of -1, so if we put 0 in the fraction for x, we get

         0+1 / 0+5 = 1/5 which is positive.

If we pick a number between -5 and -1, like -3, we get

         -3+1 / -3+5 = -2 / 2  which is negative.

If we pick -6, to the left of -5, we get

         -6+1 / -6+5 = -5 /-1  which is positive, so the sign chart looks like:

         +          -          +
         -----------|-------------|----------
         -5          -1

So the domain here would be all the values that give positive results (since you can't take a sq rt of a negative), so that would be

         (-inf, -5) U (-1, inf)

The sign chart for the second fraction under the second radical is exactly the same, since its the same quantities.

The final thing to note is that you can't allow -5 or -1 in the domain, because each makes one of the denominators = 0, so the fractions would be undefined.

So I think for the domain you want what I have above,

         (-inf, -5) U (-1, inf)


Hope this was clear enough--its hard to explain the sign charts for the first time.

Steve

---------- FOLLOW-UP ----------

QUESTION: Hi thanks for all your help on this one it was rather tricky. Would it be possibe for you to help me on this one - it does not need a domain but deals with rationalizing the denominator.

a.  1

  ____

the cube root of a + the cube root of b





b.   1
  _____

the cube root of x^2 + the cube root of xy + the cube root of y^2

Answer
Hi Mandy,

These two have to do with the formula for factoring a sum or difference of cubes:

x^3 + y^3 = (x + y)(x^2 - xy + y^2)

x^3 - y^3 = (x - y)(x^2 + xy + y^2)

In the first exercise you wrote,

1 / (cu rt a + cu rt b) , what you want is to get the radicals out of the denominator, so think of

cu rt a = x and cu rt b = y.  Then you want to multiply the top and bottom of the fraction by x^2 - xy + y^2 to get the sum of cubes:

      1          * [cu rt a^2 - cu rt a * cu rt b + cu rt b^2]
---------------------    ------------------------------------------
[cu rt a + cu rt b]     [cu rt a^2 - cu rt a * cu rt b + cu rt b^2]

= [cu rt a^2 - cu rt a * cu rt b + cu rt b^2] / [a + b}

Believe it or not, this is the rationalized form.




For the second one, think of the expression in the denominator as

 a^2 + ab + b^2   where a = cu rt x  and b = cu rt y

so, you want to multiply top and bottom by a - b to get a^3 - b^3,
which in this case would be x - y:

         1          *  [cu rt x - cu rt y}
------------------------------------         -------------------
[cu rt x^2 + cu rt xy + cu rt y^2]          [cu rt x - cu rt y]

= [cu rt x - cu rt y] / [x - y]


Hope you can understand this, its not easy with all the substitutions, but I think if you write it out step by step, you'll see what I'm getting at.

Steve