Mathematics

Problem

For a real parameter

m

, consider the rational function

p_m(x)=\frac{x^2-4}{x^2-4mx+4m^2-1}.

(a) Rewrite the denominator in completed-square form. (b) Find all real

m

for which

p_m

is defined for every real

x

. (c) When the condition in part (b) is not satisfied, (i) determine the excluded value(s) of

x

in terms of

m

, and (ii) state the domain of

p_m