Question

1. find the inverse of each relation.

a. {(3, 1), (4, -3), (8, -3)}
{(1, 3), (-3, 4), (-3, 8)}
b. {(-7, 1), (0, 5), (5, -1)}
{(1, -7), (5, 0), (-1, 5)}
c. $y = 3x - 2$

Explanation:

Step1: Swap x,y in ordered pairs

For a relation defined by ordered pairs \((x,y)\), the inverse is found by swapping the coordinates to \((y,x)\).

Part a:

Original: \(\{(3, 1), (4, -3), (8, -3)\}\)
Inverse: \(\{(1, 3), (-3, 4), (-3, 8)\}\)

Part b:

Original: \(\{(-7, 1), (0, 5), (5, -1)\}\)
Inverse: \(\{(1, -7), (5, 0), (-1, 5)\}\)

Step2: Swap x,y and solve for y

For a linear equation \(y = f(x)\), swap \(x\) and \(y\), then isolate \(y\) to find the inverse function.

Part c:

Original: \(y = 3x - 2\)
Swap \(x\) and \(y\): \(x = 3y - 2\)
Rearrange to solve for \(y\):
\(x + 2 = 3y\)
\(y = \frac{x + 2}{3}\)

Answer:

a. \(\{(1, 3), (-3, 4), (-3, 8)\}\)
b. \(\{(1, -7), (5, 0), (-1, 5)\}\)
c. \(y = \frac{x + 2}{3}\)