Download presentation

Published byYuliana Pursley Modified over 4 years ago

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 8/10/2013

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

Similar presentations

OK

5.1 Solving Systems of Equations Objectives: --To identify a system of equations --To determine if a point is a solution to a system --To use graphing.

5.1 Solving Systems of Equations Objectives: --To identify a system of equations --To determine if a point is a solution to a system --To use graphing.

© 2018 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google