The image shows two functions, $f$ and $g$, with their domains and ranges visually represented. The problem asks to determine the domain and range of the composition $g \circ f$, where the functions are composed, meaning $g(f(x))$.

Step-by-step solution:

1. Function $f$:

   • The domain of $f$: $\{1, 2, 3, 4\}$
   • The range of $f$: $\{3, 4, 5\}$
   • The mappings for $f$:
     • $f(1) = 4$
     • $f(2) = 3$
     • $f(3) = 5$
     • $f(4) = 5$

2. Function $g$:

   • The domain of $g$: $\{3, 4, 5\}$
   • The range of $g$: $\{1, 2, 6\}$
   • The mappings for $g$:
     • $g(3) = 1$
     • $g(4) = 2$
     • $g(5) = 6$

Composition $g \circ f$:

To find the composition $g \circ f$, you apply $g$ to the outputs of $f$. That is, for each $x$ in the domain of $f$, calculate $g(f(x))$:

• $g(f(1)) = g(4) = 2$
• $g(f(2)) = g(3) = 1$
• $g(f(3)) = g(5) = 6$
• $g(f(4)) = g(5) = 6$

(a) Domain of $g \circ f$:

The domain of the composition is the domain of $f$, which is: $\{1, 2, 3, 4\}$

(b) Range of $g \circ f$:

The range of the composition consists of the outputs of $g \circ f$, which are: $\{1, 2, 6\}$

Thus, the final answers are:

• Domain of $g \circ f$: $\{1, 2, 3, 4\}$
• Range of $g \circ f$: $\{1, 2, 6\}$

Would you like more details or have any questions about this? Here are some related questions for further practice:

1. How do you determine if a composition of two functions is defined?
2. Can the range of $f$ ever exceed the domain of $g$?
3. What happens if the range of $f$ has elements outside the domain of $g$?
4. How does the composition of functions relate to real-world applications?
5. What are the domain and range of the inverse of a composition?

Tip: When composing functions, always check that the output of the first function fits within the domain of the second.