Mathematics

Problem

For

a\in\mathbb R

, consider the function

F_a:\mathbb R\to\mathbb R

defined by

F_a(x)=|x|+ax

. a. (i) Rewrite

F_a(x)

without the absolute value sign by expressing it as a piecewise function. b. (ii) For a fixed

a

, determine

\displaystyle \lim_{x\to\infty}F_a(x)

and

\displaystyle \lim_{x\to -\infty}F_a(x)

. c. (iii) Using the limits found in (ii), establish all real values of

a

for which

F_a

is surjective onto

\mathbb R

. Provide full justification.