Find Inverse Function (2) - Tutorial

• Write the function as an equation.
  y = ln(x + 2) - 3
  
  
• Rewrite the equation so that it is easily solved for x.
  ln(x + 2) = y + 3
  
  
• Rewrite the above in exponential form.
  x + 2 = e^{y + 3}
  
  
• Solve for x.
  x = e^{y + 3} - 2
  
  
• now write f^-1(y) as follows .
  f ^-1(y) = e^{y + 3} - 2
  
  
  or change x into y and y into x in the above to have
  
  
  f ^-1(x) = e^{x + 3} - 2
  
  
• Check
  1. f(f ^-1(x)) = f(x) = ln(e^{x + 3} - 2 + 2) - 3
     
     
     = ln(e^{x + 3}) - 3
     
     
     = (x + 3) - 3
     
     =x
     
     
  2. f ^-1(f(x))=f ^-1(2x + 3)
     
     
     = e^{(ln(x + 2) - 3) + 3} - 2
     
     = e^{ln(x + 2)} - 2
     
     
     = x + 2 - 2
     
     = x

conclusion: The inverse of function f given above is f ^-1(x) = e^{x + 3} - 2

Matched Exercise 1: Find the inverse function of f given by