Mathematics

Problem

Let

k

be a real parameter and define

f_{k}(x)=\dfrac{\sqrt{x-2k}}{x+k}.

Answer the following, treating

k>0

k=0

and

k<0

as separate cases. (i) For a fixed value of

k

, write the inequality or exclusion that guarantees the numerator is real-valued. (ii) For the same

k

, write the exclusion that guarantees the denominator is non-zero. (iii) Combine the results of parts (i) and (ii) to obtain the domain of

f_{k}

for each of the three parameter ranges mentioned above.