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.