let f be the piece - wise function defined above. also shown is a portion of the graph of f. what is the…

Explanation:

Step1: Find left - hand limit of $f(x)$ as $x\to2$

We use the formula $f(x)=-x^{2}+3x + 3$ for $x<2$. $\lim_{x\to2^{-}}f(x)=-(2)^{2}+3\times2 + 3=-4 + 6+3=5$

Step2: Find right - hand limit of $f(x)$ as $x\to2$

We use the formula $f(x)=8-\frac{3}{2}x$ for $x > 2$. $\lim_{x\to2^{+}}f(x)=8-\frac{3}{2}\times2=8 - 3=5$ Since $\lim_{x\to2^{-}}f(x)=\lim_{x\to2^{+}}f(x)=5$, then $\lim_{x\to2}f(x)=5$

Step3: Evaluate $\lim_{x\to2}f(f(x))$

Now we need to find $f(5)$. Since $5>2$, we use the formula $f(x)=8-\frac{3}{2}x$ for $x > 2$. $f(5)=8-\frac{3}{2}\times5=8-\frac{15}{2}=\frac{16 - 15}{2}=\frac{1}{2}$

Answer:

D. $\frac{1}{2}$