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 8/10/2013

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 1-1. 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

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 1-1. 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

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. ● ● ● 8/10/2013 One-to-One Functions One-to-One Functions 8/10/2013

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

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