I have two 2-variable real functions $$-2 \;\leq \;f_1(x,y) \;\;\leq 5,$$ $$-1 \;\;\leq \;f_2(x,y)\;\; \leq 4,$$ I want to have a general condition for the range of this function $$ a_1 \;f_1(x,y)+a_2\;f_2(x,y) ,\quad \text{for} \quad a_1,a_2\in\mathbb{R}.$$ According to the inequality rules, can I claim that the following conditions hold? $$ -2\,a_1-a_2 \;\;\leq\;\; a_1 \;f_1(x,y)+a_2\;f_2(x,y) \;\;\leq \;\;\;5\,a_1+4\,a_2,\quad \text{if} \quad a_1,a_2\in\mathbb{R}^+$$ $$ -2\,a_1-a_2 \;\;\geq\;\; a_1 \;f_1(x,y)+a_2\;f_2(x,y) \;\;\geq \;\;\;5\,a_1+4\,a_2,\quad \text{if} \quad a_1,a_2\in\mathbb{R}^-$$
If so, can we say something about the situation where $\;a_1,\;a_2\;$ have different signs (one positive, one negative)?