Mathematics

Problem

For a real parameter

k

, define

f_k(x)=\sqrt{x+k}

with domain

[-k,\infty)

and

g(x)=\ln(7-2x)

with domain

(-\infty,\tfrac72)

. a. (i) Obtain an explicit algebraic expression for

(g\circ f_k)(x)

. b. (ii) Derive the domain of

(g\circ f_k)

in terms of

k

. c. (iii) Find all real values of

k

for which the domain obtained in part (ii) is non–empty. d. (iv) Determine those value(s) of

k

for which

(g\circ f_k)

is defined for every real

x\ge -k