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The image you uploaded discusses the **domain and range of composite functions**.

Here’s a breakdown of what’s covered:

1. The composite function \( g(x) = (f \circ h)(x) \) combines two functions \( f(x) \) (outside function) and \( h(x) \) (inside function).
2. **Domain and Range** relationships:
   - **h(x)** (inside function) maps values from \( X \) to \( Y \).
   - **f(x)** (outside function) takes inputs from the range of \( h(x) \), i.e., \( Y \), and outputs values in \( Z \).
   - **g(x)** (composite function) maps values from \( X \) to \( Z \).

Key points:
- The domain of \( f(x) \) (outside function) must be compatible with the range of \( h(x) \) (inside function).
- The **domain of the composite function** \( g(x) \) is the domain of \( h(x) \) because that’s where the composite function starts.
- The **range of the composite function** \( g(x) \) is the range of \( f(x) \), since the final output is determined by \( f(x) \) after applying \( h(x) \).

### The Missing Piece:
For question 3, it asks to complete the sentence: "The range of the composite function is the range of the _____ function." The correct answer would be the **outside function** \( f(x) \), as the range of the composite depends on the final transformation done by \( f(x) \).

Would you like further clarification on the composition process or any specific part?

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### Related Questions:
1. How do you determine the domain of a composite function?
2. Why must the range of the inside function \( h(x) \) fit within the domain of the outside function \( f(x) \)?
3. How would the graph of \( g(x) \) change if the range of \( h(x) \) exceeds the domain of \( f(x) \)?
4. Can you compute a composite function if the domain of \( f(x) \) is not compatible with \( h(x) \)?
5. How do you find the range of a composite function analytically?

### Tip:
Always check the range of the inside function before determining the domain of the composite function.

What are the domain and range of the composite function g(x) = (f ∘ h)(x)?

Learn how to determine the domain and range of composite functions with a step-by-step breakdown. This problem covers key concepts like function composition and how to match the domain of the inside and outside functions. Suitable for high school students, Grades 10-12.

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