ADVANCED EXERCISES-Square Roots and Other Radicals

(1) Find the domain of each function.

SOLUTION:

1. In order to calculate the square root in the function , we must have that the expression under the root sign is non-negative, so we must solve the inequality:
   Solving, we get:
   So the domain is { x | x 0 }.

2. In order to take the fourth root, the expression under the radical sign must be non-negative. So it must be true that:
   To solve this inequality, we proceed as with quadratic or polynomial inequalities. We determine the roots of the numerator and denominator. Then we determine the sign (positive or negative) of the numerator and denominator on the intervals created by the roots.

   The root of the numerator is -7, and the root of the denominator is 7, so the intervals are x < -7, -7 < x < 7, and x > 7.

   The numerator, x + 7, is negative when x < -7, and the denominator, 7 - x, is positive when x < -7. So the quotient, , is negative when x < -7.

   When -7 < x < 7, then x + 7 is positive and 7 - x is also positive. So the quotient, , is positive when -7 < x < 7.

   These results, and the case when x > 7 , are summarized in the following sign chart. (The long line with the ">" at the end represents a number line, with -7 and 7 marking off the intervals.)

        x+7            - - -              + + +              + + +
        7-x            + + +              + + +              - - -
                ___________________o___________________o___________________
                                  -7                   +7
                 - - -              + + +              - - -
   

   So we see that when -7 < x < 7, and this is the only interval for which the inequality is true.

   Additionally, since we were to solve

   
   note that when x + 7 = 0, or when x = -7.

   So the complete solution is { x | -7 x < 7 }; that is, the domain of the function is { x | -7 x < 7 }.

3. To determine the domain of the function , we look for values of t which make the expression under the radical non-negative.

   Since , we know that as well. Therefore, .

   So we know that for all t, so the domain of g(t) is all real numbers.