https://file.aiquickdraw.com/mathbot/file/1727300995028wkv8qs58.jpg

The problem involves determining the domain and range of the composition of two functions, \( f \circ g \), based on the provided diagrams.

### Understanding the Diagram:
- The first diagram shows the function \( g \), mapping values from its domain to its range.
- The second diagram shows the function \( f \), mapping values from its domain to its range.

### Approach:
To find the domain and range of \( f \circ g \), we need to understand the composition process:

1. **Composition \( f \circ g \):**
   - Apply function \( g \) first and then apply function \( f \) to the result of \( g \).
   
2. **Domain of \( f \circ g \):**
   - The domain of \( f \circ g \) consists of values in the domain of \( g \) that map through \( g \) into the domain of \( f \).

3. **Range of \( f \circ g \):**
   - The range of \( f \circ g \) consists of the values in the range of \( f \) that result from applying \( f \) to the outputs of \( g \).

### Analyzing the Diagram:
- The domain of \( g \) is \( \{ 1, 2, 4 \} \), and the range of \( g \) is \( \{ 2, 3, 8 \} \).
- The domain of \( f \) is \( \{ 0, 2, 4, 8 \} \), and the range of \( f \) is \( \{ 0, 3, 7 \} \).

#### Composition Process:
- For each element in the domain of \( g \), apply \( g \), then apply \( f \) to the result:
  - \( g(1) = 2 \), and \( f(2) = 7 \).
  - \( g(2) = 3 \), and \( f(3) = 3 \).
  - \( g(4) = 8 \), and \( f(8) = 0 \).

### Final Answer:
- **(a) Domain of \( f \circ g \):** Since all elements of the domain of \( g \) map to elements within the domain of \( f \), the domain of \( f \circ g \) is the domain of \( g \), which is \( \{ 1, 2, 4 \} \).
- **(b) Range of \( f \circ g \):** The range values from applying \( f \) to the outputs of \( g \) are \( \{ 7, 3, 0 \} \).

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

Here are 5 related questions to consider:
1. How is the domain of a composition function determined?
2. What happens if the output of \( g \) does not fall within the domain of \( f \)?
3. How does the range of \( f \circ g \) relate to the individual ranges of \( f \) and \( g \)?
4. Can a composition function have a different domain than either \( f \) or \( g \) alone?
5. How would you compose functions where \( f \) or \( g \) are not injective (one-to-one)?

**Tip:** In function composition, always check that the output of the first function (here \( g \)) falls within the domain of the second function (here \( f \)).

Find the domain and range of the composition f ∘ g using the given function mappings.

Learn how to find the domain and range of the composition f ∘ g using function mappings. This exercise covers key concepts in function composition, domain restrictions, and how to apply mappings to determine the range.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation