FINDING INVERSE OF SQUARE ROOT FUNCTION

Find the inverse of each function, and state its domain and range.

Problem 1 :

f(x) = √(x - 2), x ≥ 2

Solution:

f(x) = √(x - 2)

Replace f(x) by y.

y = √(x - 2)

Interchange x and y.

x = √(y - 2)

x² = y - 2

y = x² + 2

Replace y by f^-1(x).

f^-1(x) = x² + 2

Domain:

x ≥ 0

Range:

y ≥ 2

Problem 2 :

f(x) = -√(x + 2), x ≥ - 2

Solution:

f(x) = -√(x + 2)

Replace f(x) by y.

y = -√(x + 2)

Interchange x and y.

x = -√(y + 2)

x² = (y + 2)

y = x² - 2

Replace y by f^-1(x).

f^-1(x) = x² - 2

Domain:

x ≤ 0

Range:

y ≥ -2

Problem 3 :

f(x) = √(4 - x²), -2 ≤ x ≤ 0

Solution:

f(x) = √(4 - x²)

Replace f(x) by y.

y = √(4 - x²)

Interchange x and y.

x = √(4 - y²)

x² = 4 - y²

y² = 4 - x²

y = -√(4 - x²)

Replace y by f^-1(x).

f^-1(x) = -√(4 - x²)

Domain:

0 ≤ x ≤ 2

Range:

-2 ≤ y ≤ 0

Verify that the functions are inverses:

Problem 4 :

Solution:

f(x) = 4x² + 8

Replace f(x) by y.

y = 4x² + 8

Interchange x and y.

x = 4y² + 8

x - 8 = 4y²

Replace y by g(x).

The given functions are not inverse to each other.

Problem 5 :

f(x) = x², g(x) = √x

Solution:

f(x) = x²

Replace f(x) by y.

y = x²

Interchange x and y.

x = y²

y = √x

Replace y by g(x).

g(x) = √x

The given functions are inverse to each other.

Problem 6 :

Solution:

f(x) = 2x² - 3

Replace f(x) by y.

y = 2x² - 3

Interchange x and y.

x = 2y² - 3

2y² = x + 3

Replace y by g(x).

The given functions are not inverse to each other.