FINDING INVERSE OF SQUARE ROOT FUNCTION

To find inverse of square root function, we have to follow the steps given below. 

The given square root function will be in the form

y = a√(x - h) + k

Step 1 :

Replace x and y.

Step 2 :

Using inverse operations and using square root property, solve for y.

Step 3 :

Change y as f-1(x).

Finding domain :

How the function spread over the x-axis is domain.

Finding range :

How the function spread over the y-axis is range.

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 :

f(x)=4x2+8 g(x)=14x-8

Solution:

f(x) = 4x2 + 8

Replace f(x) by y.

y = 4x2 + 8

Interchange x and y.

x = 4y2 + 8

x - 8 = 4y2

y2=x-84y=x-84y=x-82

Replace y by g(x).

g(x)=x-82

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 :

f(x)=2x2-3 g(x)=12x+3

Solution:

f(x) = 2x2 - 3

Replace f(x) by y.

y = 2x2 - 3

Interchange x and y.

x = 2y2 - 3

2y2 = x + 3

y2=x+32y=x+32

Replace y by g(x).

g(x)=x+32

The given functions are not inverse to each other.

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