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The image contains a mathematical problem asking for the function \( f[g(x)] \), where \( f(x) = x^2 + 1 \) and \( g(x) = x - 1 \). This involves finding the composition of the functions \( f(x) \) and \( g(x) \).

### Step-by-Step Solution:

1. **Substitute \( g(x) \) into \( f(x) \):**
   - \( f[g(x)] = f(x - 1) \).

2. **Replace the \( x \) in \( f(x) \) with \( x - 1 \):**
   - \( f(x - 1) = (x - 1)^2 + 1 \).

3. **Expand the expression \( (x - 1)^2 \):**
   - \( (x - 1)^2 = x^2 - 2x + 1 \).

4. **Add 1 to the expanded expression:**
   - \( f[g(x)] = x^2 - 2x + 1 + 1 = x^2 - 2x + 2 \).

### Final Answer:
The function \( f[g(x)] = x^2 - 2x + 2 \).

Would you like more details on any part of this solution, or do you have any further questions?

Here are some related questions to expand your understanding:
1. What is the result of composing \( f(x) \) with a different function, such as \( g(x) = x^2 + 2x \)?
2. How would the composition change if \( f(x) \) were a different quadratic function?
3. Can you find the inverse function of \( f(x) \) or \( g(x) \)?
4. What happens if you compose \( g(f(x)) \) instead of \( f(g(x)) \)?
5. How does the domain and range of the composed function \( f[g(x)] \) compare to the individual functions?

**Tip:** When dealing with function composition, always substitute the entire inner function expression into the outer function's variable.

The function f[g(x)] = f(x - 1) is composed of two functions, f(x) = x^2 + 1 and g(x) = x - 1. Find the solution.

Understand how to solve the function composition f[g(x)] where f(x) is a quadratic function and g(x) is a linear function. This problem is suitable for high school students in Grades 9-12, covering key concepts in algebra and function composition.

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