HOW TO FIND INVERSE OF A FUNCTION

The following steps would be useful to find inverse of a function f(x), that is f-1(x).

Step 1 : 

Replace f(x) by y. 

Step 2 :

Interchange the variables x and y. 

Step 3 : 

Solve for y.

Step 4 :

Replace y by f-1(x). 

Example 1 :

Find the inverse of the function f(x) = x - 5.

Solution :

f(x) = x - 5

Replace f(x) by y.

y = x - 5

Interchange x and y. 

x = y - 5

Solve for y.

y = x + 5

Replace y by f-1(x).

f-1(x) = x + 5

Example 2 :

Find the inverse of the function f(x) = 3x + 5.

Solution :

f(x) = 3x + 5

Replace f(x) by y.

y = 3x + 5

Interchange x and y. 

x = 3y + 5

Solve for y.

x - 5 = 3y

y = (x - 5)/3

Replace y by f-1(x).

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

f-1 (x)  =  (x - 5)/3

Example 3 :

Find the inverse of the function f(x) = x2.

Solution :

Replace f(x) by y.

y = x2

Interchange x and y. 

x = y2

y2 = x

Solve for y.

Take square root on both sides. 

y = ±√x

Replace y by f-1(x).

f-1(x) = ±√x

Example 4 :

Find the inverse of the function f(x) = log5(x). 

Solution :

f(x) = log5(x)

Replace f(x) by y.

y = log5(x)

Interchange x and y. 

x = log5(y)

Solve for y.

y = 5x

Replace y by f-1(x).

f-1(x) = 5x

Example 5 :

Find the inverse of the function f(x) = √(x + 1). 

Solution :

f(x) = √(x + 1)

Replace f(x) by y.

y = √(x + 1)

Interchange x and y. 

x = √(y + 1)

Solve for y.

x2 = y + 1

y = x2 - 1

Replace y by f-1(x).

f-1(x) = x2 - 1

Example 6 :

Find the inverse of the function f(x) = (x + 2)/(x - 5). 

Solution :

f(x) = (x + 2)/(x - 5)

Replace f(x) by y.

y = (x + 2)/(x - 5)

Interchange x and y. 

x = (y + 2)/(y - 5)

Solve for y.

x(y - 5) = y + 2

xy - 5x = y + 2

xy - y = 5x + 2

y(x - 1) = 5x + 2

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

Replace y by f-1(x).

f-1(x) = (5x + 2)/(x - 1)

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