FINDING THE INVERSE OF A LINEAR FUNCTION

Find the inverse of the given function :

Problem 1 :

f(x) = (5/6)x - 1/3

Solution :

f(x) = (5/6)x - 1/3

Let y = f(x)

Solving for x :

y = (5/6)x - (1/3)

y + (1/3) = (5/6)x

(3y + 1)/3 = (5/6)x

(6/3)(3y + 1) = 5x

2(3y + 1) = 5x 

x = (2/5) (3y + 1)

Replacing x as f^-1(x) and x as y :

f^-1(x) = (2/5) (3x + 1)

Problem 2 :

f(x) = 2x/3

Solution :

f(x) = 2x/3

Let f(x) = y

y = 2x/3

Solving for x :

x = 3y/2

Replacing x as f^-1(x) and x as y :

f^-1(x) = 3x/2

Problem 3 :

f(x) = -5x - 5/4

Solution :

f(x) = -5x - 5/4

Let f(x) = y

y = -5x - 5/4

Solving for x :

y + 5/4 = -5x

x = (-1/5)(4y + 5)/4

x = (-1/20)(4y + 5)

x = (-y/5) + (1/4)

x = (-y/5) + (1/4)

Replacing x as f^-1(x) and x as y :

f^-1(x) = (-x/5) + (1/4)

Problem 4 :

g(x) = 6x - 5

Solution :

g(x) = 6x - 5

Let g(x) = y

y = 6x - 5

Solving for x :

6x = y + 5

x = (1/6)(y + 5)

Replacing x as g^-1(x) and x as y :

So, the inverse function is,

g^-1(x) = (1/6)x + (5/6)

Problem 5 :

g(x) = 4 + x/2

Solution :

g(x) = 4 + x/2

Let g(x) = y

y = 4 + x/2

Solving for x :

y - 4 = x/2

x = 2(y - 4)

x = 2y - 8

Replacing x as g^-1(x) and x as y :

 g^-1(x) = 2x - 8

So, the required inverse function is

g^-1(x) = 2x - 8.

Problem 6 :

g(x) = 2 + 1/3x

Solution :

g(x) = 2 + 1/3x

Let g(x) = y

y = 2 + 1/3x

Solving for x :

y - 2 = (1/3) x

3(y - 2) = x

x = 3y - 6

Replacing x as g^-1(x) and x as y :

g^-1(x) = 3x - 6

So, the required inverse function is,

g^-1(x) = 3x - 6

Problem 7 :

h(x) = 4x + 3

Solution :

h(x) = 4x + 3

Let h(x) = y

y = 4x + 3

Solving for x :

y - 3 = 4x

x = (1/4)(y - 3)

Replacing x as g^-1(x) and x as y :

 h^-1(x) = (1/4) (x - 3)

So, the required inverse function is

h^-1(x) = (x - 3)/4.

Problem 8 :

h(x) = 2x + 2

Solution :

h(x) = 2x + 2

Let h(x) = y

y = 2x + 2

Solving for x :

2x = y - 2

x = (1/2) (y - 2)

Replacing x as h^-1(x) and x as y :

h^-1(x) = (1/2) (x - 2)

So, the required inverse function is,

h^-1(x) = (1/2) (x - 2)