domain of (x/1+x)/(1+(x/1+x))

Question

domain of (x/1+x)/(1+(x/1+x))

Answer 1

Since (1 + x) is a denominator (1 + x) cannot equal zero (division by zero is undefined)

Thus x cannot equal = -1

Simplifying the complex fraction by multiplying by (1+x)/(1+x)

 

               (1+x)/(1+x) times (x/(1+x)/(1+(x/(1+x)

 

                                      x/((1+x)+x)

 

                                           x/(2x +1)

 

Now 2x + 1 cannot equal 0, thus x cannot equal -1/2

The domain in interval notation is (-infinity, -1) U (-1, -1/2) u (-1/2, infinity)  

Answer 2

The domain of a function y = f(x) is the set of all possible values of x.  

 

For the given function, 1+x ≠ 0 because division by zero is undefined.  Also, 1+(x/(1+x)) ≠ 0 for the same reason.  

 

So, x ≠ -1 and 1 + (x/(1+x)) ≠ 0

 

     x ≠ -1 and (1+x+x)/(1+x) ≠ 0

 

     x ≠ -1 and (1+2x)/(1+x) ≠ 0

 

     x ≠ -1 and x ≠ -1/2

 

Domain:  (-∞, -1) ∪ (-1,-1/2) ∪ (-1/2,∞)