(a) If $f(8) = 4$, find $f^{-1}(4)$.

Since $f$ is a one-to-one function, this means that $f^{-1}(y)$ returns the input $x$ such that $f(x) = y$.

Given that $f(8) = 4$, the inverse function returns $f^{-1}(4) = 8$.

Answer: $f^{-1}(4) = 8$.

(b) If $f^{-1}(-9) = -9$, find $f(-9)$.

From the property of inverse functions, if $f^{-1}(-9) = -9$, this means that $f(-9) = -9$, since the inverse function "undoes" the effect of $f$.

Answer: $f(-9) = -9$.

Would you like further clarification on these concepts?

Related Questions:

1. What are the conditions for a function to be one-to-one (injective)?
2. How can you verify that a function is its own inverse?
3. What are the key properties of inverse functions?
4. Can a one-to-one function have the same value for two different inputs?
5. How do you find the inverse of a given function algebraically?

Tip:

For any one-to-one function $f$, the graph of its inverse $f^{-1}$ is a reflection of the graph of $f$ across the line $y = x$.