Which function is continous?

Question

A) A(x)= x

           1+x^2

 

B) B(x)=     x

              1+x^3

Answer 1

In order for a function to be continuous, it must not have any discontinuities or gaps in x values.  We can determine this by the vertical asymptote.  The vertical asymptote is the x values where the graph gets closer to but never touches.  To find the vertical asymptote, we set the denominator of the function equal to zero.

 

1 + x² = 0

 

     x² = -1

     x = √(-1)

 

Cannot have a negative number under a square-root.  Therefore, function A does not have any real vertical asymptotes.

 

 

1 + x³ = 0

 

      x³ = -1

 

      x = -1

 

The vertical asymptote for function B is x= -1.  This function is not continuous because there is a gap at x = -1

 

 

Function A is continuous.