College Algebra

Acosta/Karwowski

Chapter 5

Inverse functions and Applications

CHAPTER 5 – SECTION 2Compositions of functions

Notation and meaning

• Combinations (f + g)(x) = f(x) + g(x) (f – g)(x) = f(x) – g(x) (f · g)(x) or (fg)(x) = f(x) · g(x) (f/g)(x) = (note: (1/g(x)) = g(x)-1

• Composition (f ∘ g) = f(g(x))

Examples

• f(x) = x + 2 g(x) = h(x) = x2 – 5x

• (f + h)(x) =• (g/f)(x) = • (f∘g)(x) =• (g∘f)(x) =• (fh)(x)=• f(x)-1 =

Numeric examples

• k(x) = 3x – 9 m(x) = (2x – 7)2 n(x) =

• Find (k + m)(3)• (mn)(-3)• (k∘n)(32)

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

•

p(x)

q(x)

CHAPTER 5 – SECTION 1Inverses 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 -6 -4 0 4 8

k(x) 8 9 10 9 8x

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

• Given a function 1) find its inverse 2) determine if the inverse is a function 3) state domain and range for each function and each inverse • f(x) = 3x – 9 g(x) = 4 – x2

• k(x) =

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 3Inverses 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:

• f(x) = 3x – 2 g(x) =

• k(x) = 4 – 9x l(x) =

• w(x) = v(x)=

• p(x) = q(x) = 8x3 + 9

Restricting domain to force inverse

• g(x) = x2 and k(x) = are not inverses because the domain and range of k(x) are restricted.

• We can make g(x) the inverse by restricting its domain and range the same way as k(x) is restricted

• given k(x) then k-1(x) = x2 with x ≥ 0• Given g(x) = x2 with x ≥ 0 g-1(x) = • with x <0 g-1(x) =

Examples: find inverses

• h(x) =

• J(x) = | x – 5| - restrict the function so that it is one to one and find its inverse function. State domain and range for the inverse function.

Assignment

• P440(1-15)• note - on 13- 15 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