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A One to One function A function is one to one if no more than one element in the domain is paired with an element in the range. Examples of one to one functions 1. A student and his or her spouse 2. A student and the college at which he or she is taking precalculus 3. A person and his or her social security number 4. A person and his or her passport number 5. Any function whose graph passes the horizontal test Examples of one to one functions Note every Uppercase number has exactly one lower case partner A A B B C C a b c Examples of one to one functions 1. f ( x) x 3 2. f (x) = 3x +5 3. f ( x) x 4. All odd functions Examples of functions that are not one to one 1. f (x) = |x| 2. f ( x) x 2 3. f ( x ) 1 x2 Examples Two function are inverse functions if the domain of one is the range of the other. 3x and x/3 are inverse function Domain Range x 3x Domain x Range x/3 -6 -2 -3 -1 0 0 3 1 -2 -6 -1 -3 0 0 1 3 6 2 2 6 9 3 3 9 The inverse of a function 1 1 f is written fof ( x) f ( f ( x)) x f 1 Inverse Functions Are f ( x) 2 x 6 2(1 x - 3) +6 2 x–6+6=x 1 (2x + 6) – 3 2 x + 3 – 3= x and 1 g ( x) x 3 2 Graphs of Inverse Functions f ( x) x 3 f ( x) 3 x f ( x) x 3 Note inverse functions are reflected about the line y = x