Mathematics

Problem

A family of rational functions is defined by

f_k(x)=\dfrac{x+1}{x^{2}+kx+1}

, where

k

is a real parameter. a. (i) Show that the denominator has real zeros if and only if

k^{2}-4\ge0

. b. (ii) Hence find all real values of

k

for which the domain of

f_k

is the entire set of real numbers. c. (iii) For

k=5

, determine the domain of

f_k

and express it in interval notation. d. (iv) For

k=-1

, find the domain and the range of

f_k