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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 = ±√(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.
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