Section 4.2 Combining Functions; Composite Functions... 1 Section 4.3 Inverse Functions... 4

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1 Property: T. Hrubik-Vulanovic Chapter 4 Additional Topics with Functions Content: Section 4.2 Combining Functions; Composite Functions... 1 Section 4. Inverse Functions... 4 Section 4.2 Combining Functions; Composite Functions When functions are combined using algebraic operations +, -, * and /), the result is a function. Example from section 1. is a profit P) represented as a difference between revenue R) and cost C). For Rx) = 2x and Cx) = 1x + 1, profit Px) is computed as: Px) = Rx) - Cx) = 2x - 1x + 1) = 2x - 1x - 1 = 1x - 1 As opposed to combined functions where each function independently works on the same domain x-value), composite functions work on the output of the previous function: ) In this case ) works on, output of function. As a result computation is done from the inside out: must be computed first and then computed value is plugged into. Example: If fx) = x and gx) = x-1, compute ) ) ) ), so the final answer is ) Order of functions often makes the difference in composite functions: not the same as ) Example: If fx) = x and gx) = x - 1, compute Class notes 2 ) is often ) Page 1 of 7

2 ) ), so the final answer is ) Example - the difference between combined and composite functions. Just read it. There is no need to put this into your notes. A person x) needs to do hairdo and manicure hairdo and manicure are two functions. When these two functions operate on the same person x), the result is manicured person with a nice hairdo r). Adding two functions makes them combined functions. Now consider washing the hair and blow-dry. These two functions done on the same person x) represent composite functions and, depending on the order, they will produce two different results: Example 7. If and gx) = 2x² - 4. c. Compute and without using calculator. ) g-)= 2*-)²-4=2*-4=18-4=14 = The answer is: ) g2)=2*2²-4=8-4=4 Class notes 2 Page 2 of 7

3 Example: If ) ) and gx) = 2x² - 4 compute ) ) )) this not defined in rational numbers. Since is not defined, ) is also not defined. ) ) is not defined. We found out that while This shows two things: 1. The order in which the functions are applied usually produces different results. 2. The domains of both and affect the result of composite functions. This is why the definition of composite functions listed below must include statement about the domains as well. The composite function ) ) is denoted by and is defined as: ) The domain of also of is the subset of the domain for for which is defined. for which is defined. ) The domain of is the subset of the domain for ICE 4.2:21 Use fx)=2x² and to evaluate composite functions. Also discuss the domains. ) a. b. ) ) So far we computed the values for composite functions but we can leave variable as is, to get a generic composite function: Example. Find ) using and h f)x) = h fx) f x )=2x-5 h 2x-5 )= Class notes 2 Page of 7

4 ) Section 4. Inverse Functions Composite functions where order does not matter and the result is identity function y=x) are inverse functions. and g for which ) and ) are called inverse functions. Functions for all x in the domain of g for all x in the domain of If that is the case, then g can be denoted as read inverse). Note that in this case -1 does not represent exponent. In other words: How do we know if a function has an inverse function If is a one-to one function then it has a.? A one-to-one function has for every element of the range only one element in the domain. Example of the function that does not have inverse function is Example of the function that has an inverse function is y=x^2 no inverse y=x^ 27 has inverse function If no horizontal line intersects the graph of a function in more than one point then the function is one-to-one function and therefore it has an inverse function). Steps to find inverse function: 1. Rewrite the equation replacing fx) with y Class notes 2 Page 4 of 7

5 2. Interchange x and y in the equation.. Solve new equation for y. If the solution is not unique then there is no inverse function. 4. Replace y with Example. a. Find the inverse function of a. Graph and its inverse function on the same axes. Rewrite the equation replacing with Interchange and in the equation Solve the new equation for Replace with y=2x-1)/ y y-inverse -1 y=x Graphs of inverse functions are symmetrical with respect to the line y=x. Inverse functions on limited domains Class notes 2 Page 5 of 7

6 Some functions may not have inverse functions when the domain are all real numbers but if the domain is a subset of where they are on-to-one function, on that limited domain they have inverse function. Example: The function The function on on ) has no inverse function. ) is one-to-one and has the inverse function. or ) or y=x^2 no inverse y=x^2 y So if the starting function is: where domain is restricted to steps to find inverse function are the same as before only we must carry on domain restriction as well. Rewrite the equation replacing with Interchange and in the equation and restriction. Solve the new equation for Replace Class notes 2 with where x> where y> but since y> final solution is and x> because of even root x> Page of 7

7 2 y=x^ y y-inverse y=x ICE: 4.:1 Find inverse function for Class notes 2 Page 7 of 7