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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: $\{(5,A), (9,K), (5,O), (9,Z)\}$

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, $f^{-1}(x) = 3x + 5$.

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 $y = x$.

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.