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 :
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.
May 21, 24 08:51 PM
May 21, 24 08:51 AM
May 20, 24 10:45 PM