Assume that the function

𝑓

is a one-to-one function.

(a) If

𝑓

(

8

)

=

4

, find

𝑓

-

1

(

4

)

.

Your answer is

(b) If

𝑓

-

1

(

-

9

)

=

-

9

, find

𝑓

(

-

9

)

.

Your answer is

### (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 \( y = x \).

Assume that the function f is a one-to-one function. (a) If f(8) = 4, find f^{-1}(4). (b) If f^{-1}(-9) = -9, find f(-9).

Learn how to solve problems involving inverse functions with a one-to-one function. This example includes finding f^{-1}(4) given f(8) = 4 and determining f(-9) given f^{-1}(-9) = -9. Suitable for students in Grades 9-11.

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