4.1 Inverse Functions

Inverse Functions Section 4.1 Identify One-to-One Functions One-to-One Function: A function f is a one-to-one function if, for a and b in the domain of f , if then ( ) ( ) a b f a f b   or equivalently if ( ) ( ) then f a f b a b   Determine if the relation defines y as a one-to-one function of x . 1.           2,1 , 4,2 , 7,3 , 2,1  2. A function ( ) y f x  is a one-to-one function if no horizontal line intersects the graph in more than one place 3. 4. 5. Use the definition of a one-to-one function to determine whether the function is one-to-one. (Show that if if ( ) ( ) then f a f b a b   ) 6. ( ) 3 2 f x x   7. 2 ( ) 3 f x x  Batman Green Lantern Wolverine Sinestro Joker Sabretooth

Determine Whether Two Functions are Inverses Inverse Functions: Let f be a one-to-one function. Then g is the inverse of f if the following conditions are both true: 1.   ( ) f g x x  for all x in the domain of g . 2.   ( ) g f x x  for all x in the domain of f . Given a function ( ) f x and its inverse 1 ( ) f x  , then the definition implies that     1 1 ( ) and ( ) f f x x f f x x     Determine whether the two functions are inverses. 8. 1 ( ) 3 6 and ( ) 2 3 f x x g x x     9. 1 ( ) and ( ) 5 1 5 x f x g x x     Find the Inverse of a Function Procedure to Find an Equation of an Inverse of a Function For a one-to-one function defined by ( ) y f x  , the equation of the inverse can be found as follows: Step 1 : Replace ( ) f x by y . Step 2 : Interchange x and y . Step 3 : Solve for y . Step 4 : Replace y by 1 ( ) f x  .

A one-to-one function is given. Write an equation for the inverse function. 10. 1 ( ) 2 2 f x x   11. 3 ( ) 1 x f x x  