To solve $f(g(x))$ given the equations:

1. $f(x) = x^2$
2. $g(x) = x - 1$

Step-by-step solution:

1. Substitute $g(x) = x - 1$ into $f(x) = x^2$: $f(g(x)) = f(x - 1)$

2. Replace $x$ in $f(x) = x^2$ with $x - 1$: $f(x - 1) = (x - 1)^2$

3. Expand $(x - 1)^2$: $(x - 1)^2 = x^2 - 2x + 1$

Thus, $f(g(x)) = x^2 - 2x + 1$.

Final Answer:

$f(g(x)) = 1x^2 - 2x + 1$

Would you like further clarification or details?

Here are 5 related questions for deeper understanding:

1. What is the general process for finding compositions of functions like $f(g(x))$?
2. How can we determine $g(f(x))$ using the same functions?
3. What is the importance of expanding polynomials like $(x-1)^2$ in function compositions?
4. How do compositions of functions apply in real-world problems?
5. Can you identify whether $f(g(x))$ is a one-to-one or onto function?

Tip: When solving function compositions, always start by substituting the inner function into the variable of the outer function!