Find the domain of the following function:

Question

         3x^(3)-pi                  1

f(x)=———————    + ———————

        sq.rt.(2x+3)-5        |x+1|-1

 

The answer I got was (-inf.,-3/2)U(-3/2,inf.)

Answer 1

The denominator cannot be zero, and values under the square-root cannot be negative.  So we set the following cases:

 

2x + 3 > 0     ,     √(2x + 3) = 5       ,        x + 1 = -1  ,        and        x + 1 = 1

 

 

If we solve for x for each case, we get

 

For the 1st square-root case

 

x > -3/2              

 

 

For the second square-root case

 

2x + 3 = 25        

 

2x = 22

 

 x = 11

 

 

For the absolute value case

 

x = -2            and            x = 0           

 

 

So your domain is all real numbers greater that -3/2 excluding zero and 11.  In interval notation,

 

(-3/2, 0)∪(0, 11)∪(11, ∞)