Let's solve this step-by-step.

1. Understanding the function $f$

The given function is: $f = \{(a,b), (b,a), (c,b)\}$ This means:

• $f(a) = b$
• $f(b) = a$
• $f(c) = b$

2. Computing $f \circ f$ (i.e., $f$ composed with itself)

The composition $f \circ f$ means applying $f$ twice. In other words, $f \circ f(x) = f(f(x))$ for each element $x \in X$.

• $f \circ f(a) = f(f(a)) = f(b) = a$
• $f \circ f(b) = f(f(b)) = f(a) = b$
• $f \circ f(c) = f(f(c)) = f(b) = a$

Thus, $f \circ f$ is: $f \circ f = \{(a,a), (b,b), (c,a)\}$

3. Computing $f \circ f \circ f$

Now, we compose $f \circ f$ with $f$ again, meaning we apply $f$ to the result of $f \circ f$.

• $f \circ f \circ f(a) = f(f \circ f(a)) = f(a) = b$
• $f \circ f \circ f(b) = f(f \circ f(b)) = f(b) = a$
• $f \circ f \circ f(c) = f(f \circ f(c)) = f(a) = b$

Thus, $f \circ f \circ f$ is: $f \circ f \circ f = \{(a,b), (b,a), (c,b)\}$ Notice that this is exactly the original function $f$. This means that $f \circ f \circ f = f$.

4. Generalizing $f^n$ (composition of $f$ with itself n times)

Since $f \circ f \circ f = f$, we can observe that:

• $f^1 = f = \{(a,b), (b,a), (c,b)\}$
• $f^2 = f \circ f = \{(a,a), (b,b), (c,a)\}$
• $f^3 = f \circ f \circ f = f = \{(a,b), (b,a), (c,b)\}$

From this, we can conclude that the composition alternates between $f$ and $f \circ f$.

5. Finding $f^9$ and $f^{623}$

Since the composition alternates between $f$ and $f \circ f$, the general pattern is:

• $f^1 = f$
• $f^2 = f \circ f$
• $f^3 = f$
• $f^4 = f \circ f$
• And so on...

Thus:

• $f^9 = f$ because 9 is odd.
• $f^{623} = f$ because 623 is odd.

Final Answers:

• $f \circ f = \{(a,a), (b,b), (c,a)\}$
• $f \circ f \circ f = \{(a,b), (b,a), (c,b)\}$
• $f^9 = \{(a,b), (b,a), (c,b)\}$
• $f^{623} = \{(a,b), (b,a), (c,b)\}$

Would you like a more detailed breakdown of any part of this solution?

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Related Questions:

1. How can I find the powers of more complex functions?
2. How does function composition work when the functions have different domains and ranges?
3. What is the difference between $f \circ f$ and $f^2$?
4. What happens when we compose a function with itself an even number of times versus an odd number of times?
5. How can I visualize function composition on a graph?

Tip:

When working with function compositions, always track the pattern for small values of $n$ first—this can often reveal periodicity or repetition that simplifies calculations for larger $n$.