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College Algebra

College Algebra. Acosta/ Karwowski. Chapter 5. Inverse functions and Applications. Compositions of functions. Chapter 5 – section 2. 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)

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College Algebra

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  1. College Algebra Acosta/Karwowski

  2. Chapter 5 Inverse functions and Applications

  3. Compositions of functions Chapter 5 – section 2

  4. 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))

  5. 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 =

  6. Numeric examples • k(x) = 3x – 9 m(x) = (2x – 7)2 n(x) = • Find (k + m)(3) • (mn)(-3) • (k∘n)(32)

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

  8. Inverses of relations Chapter 5 – Section 1

  9. 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

  10. 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(??) = ??

  11. 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.

  12. Examples: • Decide if the function is one to one

  13. 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) =

  14. 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:

  15. 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

  16. 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)

  17. Inverses defined by compositions Chapter 5 – section 3

  18. 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

  19. 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

  20. 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) =

  21. 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.

  22. 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

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