2020 Maths Advanced Sample HSC Q17 Sketch composite function graph f(g(x)), f(x)=x^2+2, g(x)=√(x-6)

Source: © NSW Education Standards Authority

Disclaimer: This sample solution is intended as a guide only and does not necessarily represent the best way to answer the question, nor is there any guarantee of accuracy.
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Sketching the graph of a composite function involves understanding how the individual functions behave and how they affect each other when combined. Here are some general steps to help you sketch the graph of a composite function:

1. **Understand the Inner Function:**
- Identify the inner function (the function inside the parentheses). Understand its basic shape and key features.

2. **Understand the Outer Function:**
- Identify the outer function (the function outside the parentheses). Again, understand its basic shape and key features.

3. **Compose the Functions:**
- Combine the inner and outer functions to form the composite function. This is done by replacing the variable in the outer function with the entire inner function.

4. **Analyse the Key Points:**
- Identify key points on the graph, such as intercepts, critical points, and asymptotes. Use these to help plot the composite function.

5. **Consider the Domain:**
- Be mindful of the domain of the composite function. Ensure that the inner function's domain is compatible with the outer function.

6. **Graph the Composite Function:**
- Plot points on the graph based on the composition of the functions. Use the information from steps 1-5 to guide your sketch.

7. **Check Symmetry and Transformations:**
- Consider any symmetry or transformations that may apply to the composite function. These can affect the overall shape and orientation of the graph.

8. **Refine the Graph:**
- Refine your sketch by connecting the points smoothly and ensuring that the graph reflects the characteristics of both functions.

9. **Label and Interpret:**
- Label important points, axes, and any other relevant details on the graph. Interpret the graph in terms of the original functions and the composition.

Remember that practice is key when it comes to sketching graphs. The more familiar you become with different types of functions and their behaviors, the more confident you'll be in sketching composite functions.