FINDING INVERSE OF QUADRATIC FUNTIONS

Problem 1 :

f(x) = x² + 6x + 4

Solution:

f(x) = x² + 6x + 4

Replace f(x) by y.

y = x² + 6x + 4

Interchange x and y.

x = y² + 6y + 4

x = (y² + 6y + 9) - 9 + 4

x = (y + 3)²  - 5

(y + 3)² = x + 5

Take square root on both sides.

y + 3 = ± √(x + 5)

y = ± √(x + 5) - 3

Replace y by f^-1(x).

f^-1(x) = ± √(x + 5) - 3

Problem 2 :

f(x) = 4x² - 8x - 5

Solution:

f(x) = 4x² - 8x - 5

Replace f(x) by y.

y = 4x² - 8x - 5

Interchange x and y.

x = 4y² - 8y - 5

= 4(y² - 2y) - 5

= 4(y² - 2y + 1 - 1) - 5

= 4[(y - 1)² - 1] - 5

= 4(y - 1)² - 4 - 5

x = 4(y - 1)² - 9

x + 9 = 4(y - 1)²

Take square root on both sides.

Replace y by f^-1(x).

Problem 3 :

f(x) = x² - 8x + 12

Solution:

f(x) = x² - 8x + 12

Replace f(x) by y.

y = x² - 8x + 12

Interchange x and y.

x = y² - 8y + 12

x = y² - 8y + 4² - 4² + 12

= (y - 4)² - 16 + 12

x = (y - 4)² - 4

x + 4 = (y - 4)²

Take square root on both sides.

y - 4 = ± √(x + 4)

y = ± √(x + 4) + 4

Replace y by f^-1(x).

f^-1(x) = ± √(x + 4) + 4

Problem 4 :

f(x) = 9x² + 12x + 7

Solution:

f(x) = 9x² + 12x + 7

Replace f(x) by y.

y = 9x² + 12x + 7

Interchange x and y.

x = 9y² + 12y + 7

Take square root on both sides.

For each of the following functions, calculate the inverse function.

Problem 5 :

f(x) = x² + 6x + 1

Solution:

f(x) = x² + 6x + 1

Replace f(x) by y.

y = x² + 6x + 1

Interchange x and y.

x = y² + 6y + 1

x = (y² + 6y + 9) - 9 + 1

= (y + 3)² - 8

x + 8 = (y + 3)²

Take square root on both sides.

y + 3 = ± √(x + 8) 

y = ± √(x + 8) - 3

Replace y by f^-1(x).

f^-1(x) = ± √(x + 8) - 3

Problem 6 :

f(x) = x² - 4x + 12

Solution:

f(x) = x² - 4x + 12

Replace f(x) by y.

y = x² - 4x + 12

Interchange x and y.

x = y² - 4y + 12

x = (y² - 4y + 4) - 4 + 12

x = (y - 2)² + 8

x - 8 = (y - 2)²

Take square root on both sides.

y - 2 = ± √(x - 8) 

y = ± √(x - 8) + 2

Replace y by f^-1(x).

 f^-1(x) = ± √(x - 8) + 2

Problem 7 :

f(x) = x² - 12x - 5

Solution:

f(x) = x² - 12x - 5

Replace f(x) by y.

y = x² - 12x - 5

Interchange x and y.

x = y² - 12y - 5

x = (y² - 12y + 36) - 36 - 5

x = (y - 6)² - 41

x + 41 = (y - 6)²

Take square root on both sides.

y - 6 = ± √(x + 41) 

y = ± √(x + 41) + 6

Replace y by f^-1(x).

f^-1(x) = ± √(x + 41) + 6

Problem 8 :

f(x) = x² + 10x + 15

Solution:

f(x) = x² + 10x + 15

Replace f(x) by y.

y = x² + 10x + 15

Interchange x and y.

x = y² + 10y + 15

x = (y² + 10y + 25) - 25 + 15

x = (y + 5)² - 10

x + 10 = (y + 5)²

Take square root on both sides.

y + 5 = ± √(x + 10) 

y = ± √(x + 10) - 5