Find the inverse of each relation or function. Determine if the inverse is also a function.

(first problem)
2x + 3y = 6




(second problem)
Y = -2x - 8




(third problem)
Y = x^2 + 1

(first problem)
2x + 3y = 6
3y = 6 - 2x
=> y = 2 - 2x/3
now swap x <-> y over to get inverse:
x = 2 - 2y/3
or
3x + 2y = 6

is a function since y = f(x) is linear or straight line, so is inverse function.


(second problem)
y = -2x - 8
y + 8 = -2x
2x = -y - 8
x = -y/2 - 4

swap x <-> y

y = -x/2 - 4

is a straight line (linear) function.


(third problem)
y = x ²+ 1

y -1 = x²
x = √(y -1)

swap over x <-> y

y = √(x -1)

is not a function since the square root has both a + and - radical
is a function if we limit range to say y = +√(x -1) only.

Note that if f(x) is one-to-one, then fˉ¹(x) exists and is a function.
To find the inverse function switch x and y and then solve for y

#1
2x + 3y = 6

switch:

2y + 3x = 6

Solve for y

2y = 6-3x

y = 3-(3/2)x

fˉ¹(x) = 3-(3/2) (x)

#2

y = -2x -8

switch

x = -2y -8

Solve for y

2y = -x-8

y = -(1/2)x - 4

fˉ¹(x) = -(1/2) (x) -4

# 3

The function is not one-to-one and so there is no inverse.