To find the domain and range of the composition $f \circ g$, we first need to understand how $f$ and $g$ are mapped.

1. Analysis of Functions $g$ and $f$:

• Function $g$ maps:

  • $3 \rightarrow 2$
  • $4 \rightarrow 1$
  • $5 \rightarrow 8$

• Function $f$ maps:

  • $1 \rightarrow 1$
  • $2 \rightarrow 0$
  • $8 \rightarrow 7$

2. Composition $f \circ g$:

To find $f(g(x))$, we apply $g$ first, then apply $f$ to the result of $g(x)$.

• $f(g(3)) = f(2) = 0$
• $f(g(4)) = f(1) = 1$
• $f(g(5)) = f(8) = 7$

3. Domain and Range of $f \circ g$:

• Domain of $f \circ g$: The domain consists of the elements from the domain of $g$ that map to elements within the domain of $f$. Since all elements in the domain of $g$ (3, 4, 5) map to elements within the domain of $f$ (1, 2, 8), the domain of $f \circ g$ is: $\{3, 4, 5\}$

• Range of $f \circ g$: This is the set of all values $f(g(x))$ as $x$ varies over the domain of $g$. From our composition, the range is: $\{0, 1, 7\}$

Final Answers:

• (a) Domain of $f \circ g$: $\{3, 4, 5\}$
• (b) Range of $f \circ g$: $\{0, 1, 7\}$

Would you like more details or have any questions about this solution?

Related Questions:

1. How do you determine the domain and range of a single function from a mapping diagram?
2. What is the composition of two functions and how is it calculated?
3. Can the range of a composite function include values outside the range of the inner function?
4. What happens to the domain and range of $f \circ g$ if $g$ or $f$ is not defined for certain values?
5. How do you find the inverse of a composite function?

Tip: Always verify that the output of the inner function falls within the domain of the outer function when computing compositions.