Consider the set of ordered pairs. Verify if it is one-to-one. If so, find its inverse.
$f=\left\{\right(2,9),(4,11),(6,15),(8,20\left)\right\}$

• A. $f^{-1}=\left\{\right(\frac{1}{9},\frac{1}{2}),(\frac{1}{11},\frac{1}{4}),(\frac{1}{15},\frac{1}{6}),(\frac{1}{20},\frac{1}{8}\left)\right\}$
• B. $f^{-1}=\left\{\right(\frac{1}{2},\frac{1}{9}),(\frac{1}{4},\frac{1}{11}),(\frac{1}{6},\frac{1}{15}),(\frac{1}{8},\frac{1}{20}\left)\right\}$
• C. $f^{-1}=\left\{\right(9,2),(11,4),(15,6),(20,8\left)\right\}$
• D. The function is not one-to-one.

Answer: (C)

Verified step by step guidance

1. Step 1: Understand what it means for a function to be one-to-one (injective). A function is one-to-one if each output (second element in the ordered pairs) corresponds to exactly one input (first element). In other words, no two different inputs map to the same output.
2. Step 2: Examine the given set of ordered pairs \(f = \{(2,9), (4,11), (6,15), (8,20)\}\). Check if any output values (9, 11, 15, 20) repeat for different inputs. Since all outputs are distinct, the function is one-to-one.
3. Step 3: To find the inverse of a one-to-one function, swap each ordered pair's elements. That means the inverse function \(f^{-1}\) will have pairs where the original outputs become inputs and the original inputs become outputs.
4. Step 4: Write the inverse set by swapping each pair: \(f^{-1} = \{(9,2), (11,4), (15,6), (20,8)\}\). This new set represents the inverse relation.
5. Step 5: Verify that the inverse is also a function by checking that each input in \(f^{-1}\) is unique. Since all first elements (9, 11, 15, 20) are distinct, \(f^{-1}\) is indeed a function.