Let's solve the problem step by step.

Part (a)

We are tasked with showing that $f(g(x)) = x^2 + 22x + 121$.

Step 1: Definitions of $f(x)$ and $g(x)$

The given functions are: $f(x) = x^2, \quad g(x) = x + 11.$

Step 2: Substituting $g(x)$ into $f(x)$

The composition $f(g(x))$ means we substitute $g(x) = x + 11$ into $f(x)$: $f(g(x)) = f(x + 11).$ Since $f(x) = x^2$, replace $x$ with $x + 11$: $f(x + 11) = (x + 11)^2.$

Step 3: Expanding $(x + 11)^2$

Using the binomial expansion formula: $(x + 11)^2 = x^2 + 2(x)(11) + 11^2 = x^2 + 22x + 121.$

Thus, we have shown: $f(g(x)) = x^2 + 22x + 121.$

Part (b)

We now solve the equation: $f(g(x)) = g(f(x)).$

Step 1: Express $f(g(x))$ and $g(f(x))$

From Part (a), we already know: $f(g(x)) = x^2 + 22x + 121.$

Now calculate $g(f(x))$. Substituting $f(x) = x^2$ into $g(x) = x + 11$, we get: $g(f(x)) = g(x^2) = x^2 + 11.$

Step 2: Set the two expressions equal to each other

Equating $f(g(x))$ and $g(f(x))$: $x^2 + 22x + 121 = x^2 + 11.$

Step 3: Simplify the equation

Cancel $x^2$ from both sides: $22x + 121 = 11.$ Solve for $x$: $22x = 11 - 121,$ $22x = -110,$ $x = -5.$

Final Answer

• (a) $f(g(x)) = x^2 + 22x + 121$ (shown as required).
• (b) The solution to $f(g(x)) = g(f(x))$ is: $x = -5.$

Would you like a more detailed explanation of any step?

Here are 5 related questions for deeper exploration:

1. What is the general process for evaluating composite functions like $f(g(x))$?
2. How does the binomial expansion formula work?
3. How would the solution change if the functions $f(x)$ and $g(x)$ were different?
4. Can composite functions have multiple solutions in equations like $f(g(x)) = g(f(x))$?
5. What are some applications of composite functions in real-world problems?

Tip: Always double-check your substitution when dealing with composite functions to avoid small mistakes!