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### Problems 1–4: Finding the inverse of relations
For these, we are given a set of ordered pairs, and to find the inverse, we simply swap the coordinates of each pair. 

#### 1. a = {(4,5), (-4,5), (2,3), (-2,3)}
- Inverse: \(\{(5,4), (5,-4), (3,2), (3,-2)\}\)

#### 2. b = {(0,-1), (2,-3), (-5,1), (-4,5)}
- Inverse: \(\{(-1,0), (-3,2), (1,-5), (5,-4)\}\)

#### 3. c = {(1,4), (2,8), (3,12), (4,16)}
- Inverse: \(\{(4,1), (8,2), (12,3), (16,4)\}\)

#### 4. d = {(A,5), (K,9), (O,5), (Z,9)}
- Inverse: \(\{(5,A), (9,K), (5,O), (9,Z)\}\)

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### Problems 5–8: Finding the inverse of functions
To find the inverse of a function, we solve for \(x\) in terms of \(y\) and then switch the roles of \(x\) and \(y\).

#### 5. \(f(x) = 5x + 10\)
1. \(y = 5x + 10\)
2. Solve for \(x\): 
   \[
   y - 10 = 5x \quad \Rightarrow \quad x = \frac{y - 10}{5}
   \]
3. Switch \(x\) and \(y\): 
   \[
   f^{-1}(x) = \frac{x - 10}{5}
   \]

#### 6. \(f(x) = 3x - 12\)
1. \(y = 3x - 12\)
2. Solve for \(x\): 
   \[
   y + 12 = 3x \quad \Rightarrow \quad x = \frac{y + 12}{3}
   \]
3. Switch \(x\) and \(y\): 
   \[
   f^{-1}(x) = \frac{x + 12}{3}
   \]

#### 7. \(4y - 7x = 5\)
Solve for \(y\) in terms of \(x\):
\[
4y = 7x + 5 \quad \Rightarrow \quad y = \frac{7x + 5}{4}
\]
Thus, \(f^{-1}(x) = \frac{7x + 5}{4}\).

#### 8. \(6x - 2y = -10\)
Solve for \(y\) in terms of \(x\):
\[
-2y = -6x - 10 \quad \Rightarrow \quad y = 3x + 5
\]
Thus, \(f^{-1}(x) = 3x + 5\).

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### Problems 9–10: Given \( f = \{(-3,4), (-2,2), (2,1), (3,-1)\} \)
To find the inverse, swap the coordinates:
- \(f^{-1} = \{(4,-3), (2,-2), (1,2), (-1,3)\}\)

For the graph, you would plot both sets of points \(f\) and \(f^{-1}\) on the same plane. \(f\) will map the \(x\)-coordinates to the \(y\)-coordinates, and \(f^{-1}\) will be the reflection of these points across the line \(y = x\).

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Would you like more details on any of these solutions or assistance with graphing?

Here are 5 related questions:
1. What are the conditions for a function to have an inverse?
2. Can a relation with repeated \(y\)-values have an inverse?
3. How do you verify if two functions are inverses of each other?
4. What is the graphical significance of a function's inverse?
5. How can you algebraically check if an inverse function is correct?

**Tip**: When graphing functions and their inverses, the line \(y = x\) acts as a mirror, meaning the function and its inverse will be symmetric about this line.

Find the inverse of the given relations and functions. Sketch the graph of f and its inverse.

Learn how to find the inverse of relations and functions, including linear equations. This covers key algebraic methods and graphical interpretation for high school students (Grades 9-12).

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