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(A) Given the function $g(x)=-|x+1|+3$. We know that the absolute value function $|x+1| \geq 0$ for all $x \in \mathbb{R}$. Multiplying by $-1$,we get

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Maximum value is $3$,minimum value does not exist.

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(A) Given the function $g(x)=-|x+1|+3$. We know that the absolute value function $|x+1| \geq 0$ for all $x \in \mathbb{R}$. Multiplying by $-1$,we get $-|x+1| \leq 0$ for all $x \in \mathbb{R}$. Adding $3$ to both sides,we get $g(x) = -|x+1|+3 \leq 3$ for all $x \in \mathbb{R}$. The maximum value of $g(x)$ is $3$,which occurs when $|x+1|=0$,i.e.,at $x=-1$. Since the range of the absolute value function $|x+1|$ is $[0, \infty)$,the range of $-|x+1|$ is $(-\infty, 0]$. Therefore,the range of $g(x) = -|x+1|+3$ is $(-\infty, 3]$. Since the function decreases without bound as $x 	o \infty$ or $x 	o -\infty$,it does not have a minimum value. Thus,the maximum value is $3$ and the minimum value does not exist.

Find the maximum and minimum values for the function given by $g(x)=-|x+1|+3$.

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