https://file.aiquickdraw.com/mathbot/file/17260717741536tu6ag33.jpg

Here are the graphs for the functions:

- \( f \circ g(x) = x^2 \) (blue curve)
- \( g \circ f(x) = x^2 - 2x + 2 \) (green curve)

These show how the two compositions behave differently, with \( f \circ g \) being a simple parabola and \( g \circ f \) shifted and stretched due to the linear term.

If you'd like further details or have more questions, feel free to ask!

Here are some questions that might help you explore further:

1. What are the key differences in the behavior of \( f \circ g \) and \( g \circ f \)?
2. How do the transformations affect the shape of \( g \circ f \) compared to \( f \circ g \)?
3. How would the graph change if \( f(x) \) or \( g(x) \) were altered?
4. Can we find the points where \( f \circ g \) and \( g \circ f \) intersect?
5. How can we express these functions in matrix form, or can we explore their numerical values for specific points?

**Tip:** Composing functions results in new behaviors that can simplify or complicate the expressions, depending on the functions involved.

Explore how to compose functions f(x) = x - 1 and g(x) = x^2 + 1. Learn how to compute f(g(x)) = x^2 and g(f(x)) = x^2 - 2x + 2, and understand the differences in their behavior through graphs and matrix representations.

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