INVERSE FUNCTIONS
By: Andreast Wahyu Sugiyarta PMNR'08
( 08301244011 )
Consider the function f={(1,2),(2,4),(3,6)}. If the components of each orderer pair are interchanged we obtain {(2,1),(4,2),(6,3)}. This new shet of the ordered pairs is called the inverse functions of f-, and is denoted by the symbol f -1. Note that f -1 is also a functions because to aech element in the domain there correspondent exactly one element in the range. This follow from the fact that to each element in the range of f there correspondens exatly one element in its domains . functions such as f that have this property are called one- to- one functions.
If a function f is defined by an equation in two variable, the invers f -1 if it exist, can be defined by interchanging the variable. Because a functions defined by a first- degree equation, y = ax +b (a ≠ 0) is a one- to- one functions (for each x there exactly one y, and for each there is exactly one x), every such linear function has an inverse.
Example:
Consider the functions f defined by y=2x +3. interchanging the variable in the equation we obtain.
x=2y +3
From which we have
y =1/2 (x – 3)
Thus if f(x) =2x +3, then f -1=1/2 (x – 3)
Not all function have an inverse function. For example, if then by interchanging the varuiable we obtain:
x = y2 – 4
y2 = x + 4
y = ± ( x + 4 ) ½
The relation defined by Equation 2 is not a function, since for each x > - 4 in yhe domain, there ae two element in the range. Hence the functions defined by Equation 1 is not a one- to- one functions.
Referensi:
Siswono. 2005. Matematika Inovatif Konsep dan Aplikasinya 2. Solo: Tiga Serangkai
If a function f is defined by an equation in two variable, the invers f -1 if it exist, can be defined by interchanging the variable. Because a functions defined by a first- degree equation, y = ax +b (a ≠ 0) is a one- to- one functions (for each x there exactly one y, and for each there is exactly one x), every such linear function has an inverse.
Example:
Consider the functions f defined by y=2x +3. interchanging the variable in the equation we obtain.
x=2y +3
From which we have
y =1/2 (x – 3)
Thus if f(x) =2x +3, then f -1=1/2 (x – 3)
Not all function have an inverse function. For example, if then by interchanging the varuiable we obtain:
x = y2 – 4
y2 = x + 4
y = ± ( x + 4 ) ½
The relation defined by Equation 2 is not a function, since for each x > - 4 in yhe domain, there ae two element in the range. Hence the functions defined by Equation 1 is not a one- to- one functions.
Referensi:
Siswono. 2005. Matematika Inovatif Konsep dan Aplikasinya 2. Solo: Tiga Serangkai
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