To determine the domain and range of the composition $f \circ g$, we must analyze how the two given functions $g$ and $f$ map their values step-by-step.

Step 1: Analyze the mappings of $g$ and $f$:

1. Function $g$:

   • Domain of $g$: $\{1, 3, 6, 7, 8\}$
   • Range of $g$: $\{0, 4, 6, 7\}$

   The mappings are:

   • $g(1) = 0$
   • $g(3) = 4$
   • $g(6) = 6$
   • $g(7) = 7$
   • $g(8) = 6$

2. Function $f$:

   • Domain of $f$: $\{0, 2, 3, 4, 7\}$
   • Range of $f$: $\{1, 2, 7\}$

   The mappings are:

   • $f(0) = 1$
   • $f(2) = 2$
   • $f(3) = 2$
   • $f(4) = 7$
   • $f(7) = 2$

Step 2: Composition $f \circ g$:

The composition $f \circ g$ means applying $g$ first, then $f$. Mathematically: $(f \circ g)(x) = f(g(x))$

1. Start with the domain of $g$ ($\{1, 3, 6, 7, 8\}$) and check which outputs of $g$ are in the domain of $f$:

   • $g(1) = 0$ → $f(0) = 1$ (Valid)
   • $g(3) = 4$ → $f(4) = 7$ (Valid)
   • $g(6) = 6$ → $6 \notin \text{domain of } f$ (Invalid)
   • $g(7) = 7$ → $f(7) = 2$ (Valid)
   • $g(8) = 6$ → $6 \notin \text{domain of } f$ (Invalid)

2. From this, the valid inputs (domain of $f \circ g$) are: $\{1, 3, 7\}$

3. Next, determine the outputs of $f \circ g$:

   • $(f \circ g)(1) = f(g(1)) = f(0) = 1$
   • $(f \circ g)(3) = f(g(3)) = f(4) = 7$
   • $(f \circ g)(7) = f(g(7)) = f(7) = 2$

   Thus, the range of $f \circ g$ is: $\{1, 7, 2\}$

Final Answers:

(a) Domain of $f \circ g$: $\{1, 3, 7\}$

(b) Range of $f \circ g$: $\{1, 7, 2\}$

Let me know if you would like further clarification or additional examples! Here are related questions:

1. How do you determine the domain and range of a composition function?
2. What happens if a value is outside the domain of $f$ in a composition?
3. Can a composition function have the same domain as its inner function $g$?
4. How do you visually represent compositions of functions?
5. Why is set notation useful for domain and range?

Tip: Always ensure the output of $g$ falls within the domain of $f$ for the composition $f \circ g$ to be valid.