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

## Example 1Find the inverse function, its domain and range, of the function given bySolution 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 2Find the inverse, its domain and range, of the function given by
- 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 3Find the inverse, its domain and range, of the function given by^{ 2} -1) ; x ≤ -1
- 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 = ±√(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]
## ExercisesFind the inverse, its domain and range, of the functions given below1. f(x) = -2 √(x + 2) - 6 2. g(x) = 2 √(x ^{ 2} - 4) + 4 ; x ≥ 2
## More References and Links to Inverse FunctionsFind 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. |