Find Inverse Of Square Root Functions

Examples, with detailed solutions, on how to find the inverse of square root functions as well as their domain and range.

Example 1

Find the inverse function, its domain and range, of the function given by

f(x) = √(x - 1)

Solution to example 1
  • Note that the given function is a square root function with domain [1 , + ∞) and range [0, +∞). We first write the given function as an equation as follows
    y = √(x - 1)
  • Square both sides of the above equation and simplify
    y 2 = (√(x - 1)) 2
    y 2 = x - 1
  • Solve for x
    x = y 2 + 1
  • Change x into y and y into x to obtain the inverse function.
    f -1(x) = y = x 2 + 1
    The domain and range of the inverse function are respectively the range and domain of the given function f. Hence
    domain and range of f -1 are given by: domain: [0,+ ∞) range: [1 , + ∞)

Example 2

Find the inverse, its domain and range, of the function given by
f(x) = √(x + 3) - 5

Solution to example 2

  • Let us first find the domain and range of the given function.
    Domain of f: (x + 3) ≥ 0 which gives x ≥ - 3 and in interval form [- 3 , + ∞)
    Range of f: [- 5 , +∞)
  • Write f as an equation.

    y = √(x + 3) - 5
    which gives √(x + 3) = 5 + y
  • Square both sides of the above equation and simplify.
    (√(x + 3)) 2 = (5 + y) 2
    (x + 3) = (5 + y) 2
  • Solve for x.
    x = (5 + y) 2 - 3
  • Interchange x and y to obtain the inverse function
    f -1(x) = y = (5 + x) 2 - 3
    The domain and range of the inverse function are respectively the range and domain of the given function f. Hence
    domain and range of f -1 are given by: domain: [- 5,+ ∞) range: [- 3 , + ∞)

Example 3

Find the inverse, its domain and range, of the function given by
f(x) = - √(x 2 -1) ; x ≤ -1

Solution to example 3

  • Function f given by the formula above is an even function and therefore not a one to one if the domain is the set R. However the domain in our case is given by x ≤ - 1 which makes the given function a one to one and therefore has inverse.
    Domain of f: (- ∞ , - 1] , given
    Range:
    For x in the domain (- ∞ , - 1] , the range of x 2 - 1 is given by [0,+∞), which gives a range of f(x) = - √(x 2 -1) in the interval (- ∞ , 0].
  • Write f as an equation, square both sides and solve for x, and find the inverse.

    y = - √(x 2 -1)
    y 2 = (- √(x 2 -1)) 2
    y 2 = x 2 -1
    x 2 = y 2 + 1
    x = ~+mn~√(y 2 + 1)
  • We now apply the domain of f given by x ≤ -1 to select one of the two solutions above. Hence
    x = - √(y 2 + 1)
  • Change x into y and y into x to obtain the inverse function.
    f-1(x) = y = - √(x 2 + 1)
    The domain and range of f -1 are respectively given by the range and domain of f found above
    Domain of f -1 is given by: [0 , + ∞) and its range is given by: (- ∞ , -1]



Exercises

Find the inverse, its domain and range, of the functions given below
1. f(x) = -2 √(x + 2) - 6
2. g(x) = 2 √(x
2 - 4) + 4 ; x ≥ 2


Answers to above exercises
1. f -1(x) = (1/4)(x + 6) 2 - 2 ; domain: (-∞ , - 6] Range: [- 2 ; ∞)
2. g -1(x) = √((y - 4) 2 / 4 + 4) ; domain: [4 , +∞) Range: [2 , +∞)

More References and Links to Inverse Functions


Find the Inverse of a Square Root Function
Find the Inverse Functions - Calculator
Applications and Use of the Inverse Functions
Find the Inverse Function - Questions
Find the Inverse Function (1) - Tutorial.
Definition of the Inverse Function - Interactive Tutorial
Find Inverse Of Cube Root Functions.
Find Inverse Of Square Root Functions.
Find Inverse Of Logarithmic Functions.
Find Inverse Of Exponential Functions.