Domain of function

Question

What is the domain for ln(x)=√5-x

 

** 5-x is all under the same root**

Answer 1

 

The domain of a function includes all values of x for which the function has a "defined" real value.

 

√(5 - x) does not have real values when (5 - x) is negative, because that would result in an imaginary (non-real) result.

 

So we need the part under the radical to always be greater than or equal to 0.

 

    5 - x ≥ 0

        -x ≥ 0 - 5

        -x ≥ -5

         x ≤ 5   .... reverse inequality when multiplying by a negative

 

Also, since you can only take natural logs of positive numbers, you must exclude 0 and all the negative numbers.

 

The domain is all values of x larger than 0 and less than or equal to 5   ... { x | 0 < x ≤ 5 }  

 

Answer 2

x must be > 0 and 5 - x must be >= 0 , therefore x must be <= 5

 

Domain of this implicit function : ]0, +5] , any positive real number less or equal to 5