1. Functions with Inverses Functions with Inverses
One-to-One Functions
Functions with Inverses
One-to-One Functions
We here take up the discussion of one-to-one functions: what they are, how they work, and conditions necessary for their existence.
One-to-One Functions
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2. Functions with Inverses
One-to-One Functions
Functions with Inverses
One-to-One Functions
Definition
A function f is a one-to-one function if no two ordered pairs of f have the same second component
Note:
One-to-one is often written as 1-1
One-to-One Functions
We lay the groundwork for the study of inverse functions by considering one-to-one functions. The definition is framed as simply as possible in the same terms as was the definition of function itself – that is, in terms of a set of ordered pairs.
While function is defined as a set of ordered pairs no two of which have the same first component, a one-to-one (1-1) function is first of all a function, but one in which no two ordered pairs have the same second component. Thus, a 1-1 function is a set of ordered pairs no two of which have either the same first or second components.
The examples show functions which are 1-1, as well as some that are not Some are shown as sets of ordered pairs (with no formula) and some defined by algebraic expressions (or formulas).
One example shows a set of ordered pairs which is not a function, since it contains the distinct pairs (5, 3) and (5, 6) having the same first component. This would say that the functional value of 5 is 3, but also that the functional value of 5 is 6. If the function is called f, then f(5) = 3 and f(5) = 6. Since
f(5) = 3 ≠ 6 = f(5) we then have the awkward condition that f(5) ≠ f(5).
8/10/2013
One-to-One Functions
One-to-One Functions
8/10/2013

3. Functions with Inverses
One-to-One Functions
Functions with Inverses
One-to-One Functions
1-1 Examples:
1. f = { (1, 3), (2, 5), (3, 2), (7, 1) }
2. g = { (1, 3), (2, 5), (3, 6), (7, 3) }
3. h = { (5, 3), (2, 9), (5, 6), (8, 7) }
4. f(x) = (x – 4)2 + 7
5. g(x) = x + 1
6. f(x) = |x + 1|
NOT 1-1
NOT a function
NOT 1-1
WHY ?
One-to-One Functions
We lay the groundwork for the study of inverse functions by considering one-to-one functions. The definition is framed as simply as possible in the same terms as was the definition of function itself – that is, in terms of a set of ordered pairs.
While function is defined as a set of ordered pairs no two of which have the same first component, a one-to-one (1-1) function is first of all a function, but one in which no two ordered pairs have the same second component. Thus, a 1-1 function is a set of ordered pairs no two of which have either the same first or second components.
The examples show functions which are 1-1, as well as some that are not Some are shown as sets of ordered pairs (with no formula) and some defined by algebraic expressions (or formulas).
One example shows a set of ordered pairs which is not a function, since it contains the distinct pairs (5, 3) and (5, 6) having the same first component. This would say that the functional value of 5 is 3, but also that the functional value of 5 is 6. If the function is called f, then f(5) = 3 and f(5) = 6. Since
f(5) = 3 ≠ 6 = f(5) we then have the awkward condition that f(5) ≠ f(5).
NOT 1-1
WHY ?
8/10/2013
One-to-One Functions
One-to-One Functions
8/10/2013

4. Functions with Inverses
One-to-One Functions
Functions with Inverses
Horizontal Line Test
No horizontal line intersects the graph of a 1-1 function more than once
Examples x
y(x) x
y(x)
1-1 function
Not 1-1
Horizontal Line Test
Graphical interpretation of the 1-1 property can be done with the horizontal line test, analogous to the vertical line test for the graph of a function. The examples start with the graph of a 1-1 function, followed by the graphs of two functions that are not 1-1.
The graph in the lower center passes the horizontal line test, but NOT the vertical line test – that is, the graph does not even represent a function and hence cannot be a 1-1 function.
The last graph is clearly the graph of function, but is not 1-1 as shown by the horizontal line test. ● ● ●
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One-to-One Functions
One-to-One Functions
8/10/2013

5. Functions with Inverses
One-to-One Functions
Functions with Inverses
Horizontal Line Test
No horizontal line intersects the graph of a 1-1 function more than once
More Examples x
y(x) ● ● x
y(x) x
y(x)
Horizontal Line Test
Graphical interpretation of the 1-1 property can be done with the horizontal line test, analogous to the vertical line test for the graph of a function. The examples start with the graph of a 1-1 function, followed by the graphs of two functions that are not 1-1.
The graph in the lower center passes the horizontal line test, but NOT the vertical line test – that is, the graph does not even represent a function and hence cannot be a 1-1 function.
The last graph is clearly the graph of function, but is not 1-1 as shown by the horizontal line test.
Not 1-1 ● ●
Not 1-1 ● ● ● ●
NOT a function !
8/10/2013
One-to-One Functions
One-to-One Functions
8/10/2013

6. Functions with Inverses
Think about it !
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One-to-One Functions
One-to-One Functions
8/10/2013