Maximum value of ƒ(x)

$\forall -\dfrac 2 3 \leq x, ~\min\left(2x+2,~\dfrac 1 2 x + 1, ~\dfrac 3 4 x + 7\right) = \dfrac x 2 + 1 \\ \forall x < -\dfrac 2 3, ~\min\left(2x+2,~\dfrac 1 2 x + 1, ~\dfrac 3 4 x + 7\right) = 2x+2 \\ x \to \infty \Rightarrow \dfrac x 2 + 1 \to \infty \\ \text{and thus the Min of these three functions is unbounded.}\\ \text{One might say it's maximum values is }\infty \\ \text{but it's probably better to just say it has no maximum value}$