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One-to-One Functions;
Section 6.2 One-to-One Functions; Inverse Functions 1
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Relations Definition: A relation is a set of points, (ordered
pairs) in the plane. As an example, consider the relation R {(2 , 1), (4 , 3), (0 , 3 )} As written, R is described using the roster method. Since R consists of points in the plane, we follow our instinct and plot the points.
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Relations Doing so produces the graph of R.
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Inverse Relation Definition: If R is a relation, then the relation
R-1 {(y , x) | (x , y) R } is called the inverse relation of R. The inverse relation R-1 is obtained from R by interchanging the x and y for every point in R.
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Inverse Relation - Example
As an example, consider the relation R {(2 , 1), (4 , 3), (0 , 3 )} The inverse relation R-1 is given by R-1 {(1, 2 ), (3 , 4), (3 , 0 )} Notice that the graph of R-1 is obtained from the graph of R by reflecting about the line y x all the points in R.
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Inverse Relation - Example
The graph of R and of R-1 (in red).
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Domain and Range of R-1 Definition: If R is relation, then the relation R-1 {(y , x) | (x , y) R } is called the inverse relation of R. From the definition follows that, Dom R-1 Ran R Ran R-1 Dom R
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R-1 When R is a Function If a relation f is a function defined by an equation of the form y f (x), that is, if f {(x , y) | y f (x) } then, the inverse relation f -1 of f is defined by the new equation x f (y), that is, f -1 {(x , y) | x f (y) }
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R-1 When R is a Function In this case, the fact that f is a function does not automatically imply that f -1 is also a function. For instance, f {(x , y) | y x2 } is a function but f -1 {(x , y) | x y2 } is not.
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R-1 When R is a Function f {(x , y) | y x2 } f -1 {(x , y) | x y2 }
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One to One Function The previous discussion leads to the following
definition: This condition guarantees that the inverse relation f-1 of a one to one function f is also a function. In this case, f-1 is called inverse function of f.
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Examples
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For each function, use the graph to determine whether the function is one-to-one.
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Theorem A function that is increasing on an interval I is a one-to-one function in I. A function that is decreasing on an interval I is a one-to-one function on I.
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To find the inverse we interchange the elements of the domain with the elements of the range.
The domain of the inverse function is {0.8, 5.8, 6.1, 6.2, 8.3} The range of the inverse function is {Indiana, Washington, South Dakota, North Carolina, Tennessee}
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Find the inverse of the following one-to-one function:
{(-5,1),(3,3),(0,0), (2,-4), (7, -8)} State the domain and range of the function and its inverse. The inverse is found by interchanging the entries in each ordered pair: {(1,-5),(3,3),(0,0), (-4,2), (-8,7)} The domain of the function is {-5, 0, 2, 3, 7} The range of the function is {-8, -4,0 ,1, 3). This is also the domain of the inverse function. The range of the inverse function is {-5, 0, 2, 3, 7}
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