### Related Formula f(g(x)) = f(y) quad textwhere y = g(x)$f(g(x)) = f(y) \quad \text{where } y = g(x)$ The domain restrictions of f(y)$f(y)$ must be cross-checked against the range evaluated for g(x)$g(x)$. ### Core Logic First, analyze the range of the inner function g(x)$g(x)$: g(x) = -x$g(x) = -x$ for -3 le x le 0$-3 \le x \le 0$. The range here is [0, 3]$[0, 3]$. g(x) = x$g(x) = x$ for 0 lt x le 1$0 \lt x \le 1$. The range here is (0, 1]$(0, 1]$. Therefore, the complete range of g(x)$g(x)$ across its entire domain [-3, 1]$[-3, 1]$ is [0, 3]$[0, 3]$. Note that g(x) ge 0$g(x) \ge 0$ for all valid x$x$.

Now, plug g(x)$g(x)$ into f(x)$f(x)$: f(g(x)) = begincases2+2g(x),& -1 le g(x) lt 0 quad dots (1) \\ 1-fracg(x)3,& 0 le g(x) le 3 quad dots (2)endcases$f(g(x)) = \begin{cases}2+2g(x),& -1 \le g(x) \lt 0 \quad \dots (1) \\ 1-\frac{g(x)}{3},& 0 \le g(x) \le 3 \quad \dots (2)\end{cases}$

### Step 1: Evaluate the Branches Check branch (1): -1 le g(x) lt 0$-1 \le g(x) \lt 0$. Since we already established the range of g(x)$g(x)$ is entirely non-negative (g(x) ge 0$g(x) \ge 0$), this branch condition is never satisfied. No values of x$x$ map here (x in phi$x \in \phi$). Check branch (2): 0 le g(x) le 3$0 \le g(x) \le 3$. This perfectly aligns with the entire range of g(x)$g(x)$. Hence, this branch is active for all x in [-3, 1]$x \in [-3, 1]$. We evaluate f(g(x)) = 1 - fracg(x)3$f(g(x)) = 1 - \frac{g(x)}{3}$. Since g(x)$g(x)$ spans all values continuously from 0$0$ to 3$3$: When g(x) = 0 Rightarrow f(g(x)) = 1 - 0 = 1$g(x) = 0 \Rightarrow f(g(x)) = 1 - 0 = 1$. When g(x) = 3 Rightarrow f(g(x)) = 1 - 1 = 0$g(x) = 3 \Rightarrow f(g(x)) = 1 - 1 = 0$. Since g(x)$g(x)$ is continuous and covers [0, 3]$[0, 3]$, 1 - fracg(x)3$1 - \frac{g(x)}{3}$ covers [0, 1]$[0, 1]$ continuously. ### Step 2: Final Conclusion The range of f(g(x))$f(g(x))$ is purely evaluated from branch (2), leading to [0, 1]$[0, 1]$. ### Pattern Recognition When finding the range of f(g(x))$f(g(x))$, evaluate the global range of g(x)$g(x)$ first. Use that resulting interval as the "domain" input for f(x)$f(x)$ to trace the final output bounds, eliminating unneeded piecewise branches. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 11 Mathematics: Sets, Relations and Functions