College Algebra Acosta/Karwowski

Chapter 5 Inverse functions and Applications

CHAPTER 5 – SECTION 2 Compositions of functions

Notation and meaning

Examples

Numeric examples

Examples reading graph find (p + q)(5) (pq)(-3) (q ∘p)(-6) p(x) q(x)

CHAPTER 5 – SECTION 1 Inverses of relations

Definition of inverse Inverse operators: add/subtract mult/div power/root Inverse numbers : 2 and -2 are additive inverses 2 and ½ are multiplicative inverses Generalizing - inverses “cancel” - return you to the original condition Functions – input gives output --- inverse of function – output returns the same # that you input (returns you to the original number) domain and range are interchanged – this sometimes IMPOSES a restriction on the domain of the inverse of the function - notation f -1 (x) means: the inverse of function f NOTE: the inverse of a function is NOT always a function

Examples of finite functions and their inverses f(x) =y is given to be {(2,4)(3,7)(4,13)(5,10)} If m(2) = 5 Then f -1 (x) is: Then m -1 (??) = ?? x k(x) x k -1 (x)

One- to – one function A function is one to one if there is exactly one INPUT matched to each output If the y value does not repeat If you can solve for x and get only one answer If the graph passes the horizontal line test (it is strictly increasing or strictly decreasing) If its inverse is also a function.

Examples: Decide if the function is one to one

Inverse equations

Inverses from graphs Choose some key points (like transformations) Switch the (x,y) to (y,x) graph the new points The graph is a reflection across the diagonal line y = x Ex:

Summary Inverse of function: Exchanges domain and range Switches the order of ordered pairs “flips” the graph across the y = x diagonal line Is not always a function “solves” for x

Assignment Odd problems P412(1-19) directions - for each function: a. determine if it is one to one b. determine its domain c. determine its range d. determine its inverse e. state the domain and range of the inverse f. state whether the inverse is a function or not g. determine whether the inverse is one to one or not (21-35) find the inverse of the function (37 – 47) sketch the inverse of the function (49 – 52) – all – find the inverse of the function – state any restrictions that need to be imposed on the inverse. (51-61)

CHAPTER 5 – SECTION 3 Inverses defined by compositions

Definition f(x) and g(x) are inverses if and only if: 1. the domain of f is the same as the range of g 2. the range of “f” is the same as the domain of “g” 3. (f ∘ g)(x) =(g ∘ f)(x) = x Since we can always make the domain and range match by restricting ourselves to a stated domain we are concerned with # 3 primarily note: (f -1 ∘ f)(x) =x for ALL numbers

Prove that the functions are inverses:

Restricting domain to force inverse

Examples: find inverses

Assignment P440(1-15) note - on part b and c say to “plot the points” this is confusing since they are referring to a single point.- ignore this part of the question – simply find the indicated point