Mathematics

Problem

For each real number

k

, consider the function

g_k:\mathbb R\longrightarrow \mathbb R, \qquad g_k(x)=\begin{cases}kx+2, & x\le 1,\\ x^{2}+(k-1), & x>1.\end{cases}

Determine all real values of

k

for which

g_k

is injective on

\mathbb R

, giving complete justification. a. Determine all real values of

k

for which

g_k

is injective on

\mathbb R

, giving complete justification.