Quiz (Inverse Functions)

Show all work and justify each step.

(1) Determine whether the following functions are one-to-one.

    (a)
    
    (b)
    
    (c)
    
    (d)

(2) Determine whether or not the given functions are inverses of each other or not.

    (a)   and   
    
    (b)    and   
    
(3) Find the inverse of the given function, it if exists.

    (a)
    
    (b)
    
    (c)
    

Solutions

Solutions (Inverse Functions)

(1) Determine whether the following functions are one-to-one.

    (a)
    
    Solution. The function is not one-to-one since the graph does not pass the horizontal line test as shown:
    


    
    (b)
    
    Solution. The function is one-to-one since the graph passes the horizontal line test as shown:
    


    
    (c)
    
    Solution. The function is one-to-one since the graph passes the horizontal line test as shown:
    


    
    
    (d)
    
    Solution. The function is one-to-one since the graph passes the horizontal line test as shown:
    


    

(2) Determine whether or not the given functions are inverses of each other or not.

    (a)   and   
    
    Solution. Since



and


    
we see that and are inverse functions.


    
    (b)    and   
    
    Solution. Since



and


    
we see that and are inverse functions.

    
    
    
(3) Find the inverse of the given function, if it exists.

    (a)
        
    Solution. The inverse function is found by:
    


Therefore,
    
    
    
    (b)
    
    Solution. The inverse function is found by:
    


Therefore,
    
        
    (c)
    
    Solution. The inverse function is found by:
            


and since we choose since for the inverse function we must have

Therefore,
        

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