Question

the table above gives values for the invertible function (y = f(x)) for selected values of (x).

which of the following input-output pairs describes (f^{-1})?

• option 1:

input values: (-6), (2), (3), (5)
output values: (-1/4), (1/3), (-1/7), (-1)

• option 2:

input values: (-6), (2), (3), (5)
output values: (-4), (3), (-7), (-1)

• option 3:

input values: (-6), (2), (3), (5)
output values: (4), (-3), (7), (1)

• option 4:

input values: (-3), (1), (4), (7)
output values: (1/2), (1/5), (-1/6), (1/3)

Explanation:

🆕 New Concept Discovered: Inverse Functions
Swapping inputs and outputs to reverse a function's action.

Step 1: Understand the definition of an inverse function

An inverse function, denoted as \( f^{-1} \), reverses the action of the original function \( f \).

If the original function maps an input \( x \) to an output \( y \):

$$ f(x) = y $$

Then the inverse function maps that output \( y \) back to the original input \( x \):

$$ f^{-1}(y) = x $$

In terms of tables, this means we swap the roles of the inputs and outputs:

• The outputs (\( y \)-values) of \( f \) become the inputs of \( f^{-1} \).
• The inputs (\( x \)-values) of \( f \) become the outputs of \( f^{-1} \).

---

Step 2: Swap the table values

Let's look at the given coordinate pairs \( (x, y) \) for \( f(x) \):

• \( (-3, 2) \implies f(-3) = 2 \)
• \( (1, 5) \implies f(1) = 5 \)
• \( (4, -6) \implies f(4) = -6 \)
• \( (7, 3) \implies f(7) = 3 \)

Now, we swap the coordinates to find the corresponding pairs \( (y, x) \) for \( f^{-1} \):

• \( f^{-1}(2) = -3 \)
• \( f^{-1}(5) = 1 \)
• \( f^{-1}(-6) = 4 \)
• \( f^{-1}(3) = 7 \)

---

Step 3: Match with the given options

Let's organize our new pairs into a table where the inputs are sorted in ascending order:

$$
LATEXBLOCK0
$$

This matches the third option.

Answer:

The third option is correct:

• Input values: \(-6\), \(2\), \(3\), \(5\)
• Output values: \(4\), \(-3\), \(7\), \(1\)