State the implied domain of f(x)=1/v(2x+5)-8 using interval notation

Question

State the implied domain of f(x)=1/√(2x+5)-8 using interval notation

Answer 1

The domain the set of x values where the graph exists.  In this case, f(x) is a rational function, which means that we have a numerator and a denominator.  We know that if we set the denominator equal to zero, the function is undefined.  We also have a square-root.  The number under the square-root cannot be negative, otherwise the value will not be a real number.

 

So we have to set up our restrictions.  The first one is the denominator, and the second one is the square-root.

 

√(2x + 5) - 8 = 0                 and              2x + 5 ≥ 0

 

√(2x + 5) = 8                      and              2x ≥ -5

 

2x + 5 = 64                        and                 x ≥ -5/2

 

2x = 59                               and                x ≥ -2.5

 

  x = 29.5                            and                x ≥ -2.5

 

 

This means that 29.5 and any value less than -2.5 are not in the domain.

 

The domain is  [-2.5 , 29.5)∪(29.5 , ∞).