Examples with Detailed Solutions

Example 1

• Note that the given function is a logarithmic function with domain (2 , + ∞) and range (-∞, +∞). We first write the function as an equation as follows
  y = Ln(x - 2)
  
• Rewrite the above equation in exponential form as follows
  x - 2 = e ^y
  
• Solve for x
  x = 2 + e ^y
  
• Change x into y and y into x to obtain the inverse function.
  f ^-1(x) = y = 2 + e ^x
  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: (- ∞,+ ∞) range: (2 , + ∞)

Example 2

Solution to example 2

• Let us first find the domain and range of the given function.
  Domain of f: 4 x - 6 > 0 or x > 3 / 2 and in interval form (3 / 2 , + ∞)
  Range of f: (-∞,+∞)
  
• Write f as an equation, change from logarithmic to exponential form.
  
  y = 3 Ln( 4 x - 6) - 2
  which gives Ln( 4 x - 6) = (y + 2) / 3
• Change from logarithmic to exponential form.
  4x - 6 = e ^{(y + 2) / 3}
  
• Solve for x.
  
  4x = e ^{(y + 2) / 3} + 6
  and finally x = (1/4) e ^{(y + 2) / 3} + 3/2
  
• Change x into y and y into x to obtain the inverse function.
  f^-1(x) = y = (1/4) e ^{(x + 2) / 3} + 3/2
  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: (-∞ , + ∞) and its range is given by: (3 / 2 , + ∞)

Example 3

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 R. However the domain in our case is given by x < -2 which makes the given function a one to one function and therefore has inverse.
  Domain of f: (- ∞ , -2) , given
  Range: for x in the domain (- ∞ , -2) , the range of x ² - 4 is given by (0,+∞). Since the range of the argument x ² - 4 of ln is given by (0 , +∞), the range of ln(x ² - 4) is given by (-∞, +∞) which is also the range of the given function.
  
• Find the inverse of f, write f as an equation and solve for x.
  
  y = - ln(x ² - 4) - 5
  ln(x ² - 4) = - y - 5
  Rewrite the above in exponential form
  x ² - 4 = e^{-y - 5}
  and finally x = ~+mn~ √(e^{-y - 5} + 4)
  Since x < -2 (given domain), we have x = - √(e ^{-y - 5} + 4)
  
• Change x into y and y into x to obtain the inverse function.
  f^-1(x) = y = - √(e ^{-y - 5} + 4)
  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: (-∞ , + ∞) and its range is given by: (- ∞ , -2)

Exercises

Answers to above exercises
1. f ^-1(x) = - e^{- x - 6} + 4 ; domain: (-∞ , +∞) Range: (-∞ , 4)
2. g ^-1(x) = √(1 + e ^{x + 3}) ; domain: (-∞ , +∞) Range: (1 , +∞)

More References and Links to Inverse Functions