COMPOSITION OF FUNCTIONS d INVERSES OF FUNCTIONS AND

Transcription

1 NAME UNITS ALGEBRA II COMPOSITION OF FUNCTIONS d INVERSES OF FUNCTIONS AND LOGARITHMIC d EXPONENTIALFUNCTIONS

2 Composition of Functions Worksheet Name -3O I. Let f (x) = 2x -, g(x) = 3x, and h(x) = x2 +. Compute the following:. f(x) + g(x) 2. g(x) - h(x) 0?>H + 3* "5x^ 5. g(f(0)) 6. (hog)(x) 7. f(9(h(2))) g(f(h(-6))) a («)" * 3(73"),

3 II. Let f(x) = 9 - x, g(x) = x2 + x, and h(x) = x - 2. Compute the following: 0. (go/)(3). f(g(x)) 2. h(f(-6)) K, 3. f(h(x» 4. (/2og)(H) 5. g(h(x)) QE 6. g(h(f(5))) 7. h(g(f(x))) 8. --% f fto) * <M-

4 Composition of Functions Worksheet #2 Name <,& I. Let f(x) = 3x+2, g(x) = -4x, and h(x) = x - 4. Compute the following 2. g(x) - h(x) 4. g(x) 5. g(f(0)) 6. afv) f(g(h(2») 9. 9(f(h(-6»)

5 II. Let f (x) = 7 - x, g(x) = x - x, and h(x) = x + 4. Compute the following: 0. a f(g(x)) f 2. h(f(-6)) 3. f(h(x)) 4. (hog)(n) 5. g(h(x)) [3 uo - X - 6. g(h(f(5))) 7. h(g(f(x))) 8. v\0 ^

6 Inverses of Quadratics Name: Show all work for the following on your own paper.. Given f(x) = 2(*-3)2-7 \ 'X [(\^ a. Graph f(x) on the axes to the right. ) b. State the range off(x). 4 i -5 ^-* -7-5 *!L I i -5 A n ~ _i -5 3 s c. Find/" fo) algebraically, in terms of x. d. Graph/" fjcj on the same axis e. State the domain and range off'l(x). * vu *- x^^ f. Confirm that/f^) anq/" ^cx are inverses by doing their compositions. < ^ g: Without doing any algebra, explain two different reasons you can tell you answei to pait b. is

7 2. Given f(x)- (*~2) +6 ^u-biaw a function fkm dim I fop. b. Fiadf'l(x) in terms of A:. c. Without graphing, state the domain and range off" (x) d. Evaluate /(6). - e. Evaluate /"'(-2). f. How can you tell that your answer to part b. is correct using part d. and e.?

8 Functions & Inverses Name: Let f(x) = x + 3 and g(x) = 2x2 + 6x. Perform the following function operations and simplify. 5. («>«)(-!) -_ -q ^ 4. 9 fov

9 7. Given f(x) = V* - + 3, find 5 \> "Z UN n cr b) Range off- /I c- U 3s ^ ^ \U f f X ' o -I 3^ -i *^ 2- a M r fe ** ^x^3? aj^t 9. Graph the inverse relation of the given function and determine if it is a function. Jx I v > \) Domain ->. off- <L A * -_} ^--^ / n ^ ^ A X" 3 k ^\ / V, / r ^ I ',. -^~} "* Is the inverse relation a function? Mb d) Domain of/ = (_ e) Range of/'= ^ f) Whether/ is a function or not: 8. The function g(x) = -x is its own inverse. Explain why this is true. "2 3 x 5 :) = x-\ 2 2 use composition to determine if/and g are inverses. -f -- a 0. Given f(x) = 2x-5 and =,, i LOO, 6 ly-l

10 Pre-Quiz Function and Composition Name A# f(x) = x2 Ix g(x) = 4x /i(x) = V* - 5 /c(x) = 7x - 5 Perform the following function operations and simplify given the following functions:. (/-<?)(*) 2. (/c+ #)(x) 3. ( Use composition to determine if /(x) and g(x) are inverses. Show your algebra 7. Given /(Y) = 5x 7 ^(x) Mo Find the inverse of the following functions 8. /(x) = V^7 9. = 3(x - 7)2 + 5 r- 0. /(x) = 9x - 0 x-_^-io x y+ie =?x^ «} p-

11 . Given f(x) = -3x - 5 Graph and label/(x) and/'^x) on the same axes. /N, ^. I - i L. e 2. Given the graph of/(x) below, graph /^(x) on the same axes * M

12 Review Inverses Name x X 0 i 2 y 0 i jj Given the function a. Algebraically find the inverse, /"' (x), in terms of x. *.. d b. Graph /"'(*) that you found above. c. Reflect the graph of /"' (x) over the line y = x. d. Does your reflection fall on the graph of /(*) = 2x2-5? (If not, find your mistake!) e. Show that /(x) and /"' (*)are inverses by composition. Remember, if two functions are inverses, then f /" -, On the given gri a. Graph y f(x) ^A^ = fv-2)3 ^. ^.y b. Graph /"'(*). c. Find f~l(x) in terms of A:. D O d. Show that /(x) and /"' (jc) are inverses by

13 X -3 _ i i 3. -I o 3 o the given grktr a. GrapW f(x) - 2-Jx + l - 3. b. Graph /''(*). ^ c. Find /" (x) in terms of A:. XI V -{* -7 f / \ v 4-^ \v = / Sr ) ~ (X; L a -/ 3.(x^JJ 2 3 :> J -^» ^- i i *. J 7^ ^ d. Evaluate /(3). '3 =. ^ f. How should you have known the answer to e. without doing any work? ^iu>j g. State the following: ' i. Domain off. ii. Range of/' " Domain of ': < i?- Rangeof/-: (, \l

14 y Exponential Graphs without a Calculator For each problem, Graph and state the requested information. Name:. y = y Complete the table of values, then graph. x y a. 4 J. V (\ J Y <t ^ ' ' : 37 «i o j j T I t \i.v-inte cept: t\t}f\, \R_F-mtercept: /^..\O D: R: U >O Asymptote(s): Increasing oo -^-^ <, oc? Decreasing : nit*. As x > -oo, the graph approaches 2. Using transformations, graph y = 3 *+3 i/ ^. v. 3. Using transformations, graph y = 3 - A i ^ \K >>-intercept:_ [\ "'O D R: As>ympto(fe(s): u -D In :reasing: oo^- x < '"^ D( xreasing: /vl & As x > oo, the graph approaches 4. ^ a M / Ii r Using transforma tion' ^graph y = 3V + j i \ ' / 4- t ~x~ ^-intercept: y-intercept:(0^7) D: lg_ R: Asymptoie(s): s _ Increasing: _ «V; <x-o Decreasing: As x» oo, the graph approaches As x > -oo, the graph approaches Q ^-intercept: > t ^-intercept: R:_ Asymptc>fe(s): Increasing: _ Decreasing: As x -> -oo, the graph approaches t*

15 .^, 5. Using transformations, graph y = 3x Using transfrmations, graph _y = 3 \ k G \ 5 V ---3.x-intercept: C \ J y-intercept:_ D: g. R: H > -3 Asymptote(s): U - "^y Increasing: _ --o>c^t- x <- o*^ Decreasing: j/v.j'^- As x -> -oo, the graph approaches - x-\: ntercept: MA* > r y- intercept: (O R: U 70 Asympto'ti s): M '0 In :reasing: htl^v D«:creasing _ -r^> -_ -* <. c^= As x > oo, the graph approaches As x» -oo, the graph approaches O 6. Using transforma I ] / " C 8. Using transforma :ions, graph y - -3 *. /(IX*/ i L.. _, \, t graph y = 3* - ^-intercept: y-intercept: D: ff R: -V s): ^ *-l Increasing: _ Decreasing: _ As x >oo, the graph approaches _ As x > -oo, the graph approaches ^-intercept: \ Intercept: R: Asymptote 4 *0 Increasing: Decreasing: _ n As x -> -co, the graph approaches

16 Base 3 Logarithmic Graphs without a Calculator For each problem, graph and state the requested information. Name:. Graph y = 3 Graph the line y = x. Graph the inverse of y - 3* by reflecting ij^graphjwer the line y=x.. (i-3 - p^.-^., J J / J / q, -* ^-- ' sformations, grap j i v=>l g3* + b / ei/ x x* anc *f NJ.^ -^ n*- aw.-. You just gi aphea y - log 3 x Vne inverse of y-y. *-- Complete the table of values and information below for y = Iog3*. X y - V_/v 0 -^ /9 a /3 - I E Q 3 I 9 Z* i = Iog3(x- I) J.- - in * ^ og3 x + \, find w? following. /_l \j.oj ^-intercept: D: Ixl T> 0 R: *MU Asymptc nptote(s): Increasing: Decreasing: As x t -oo, the graph approaches As x > 0, the graph approaches _ 4. Using transformations, graph y = log 3 ( Also find the equation oft! x-intercept: (J D: X j X - ) find the following. pjo) y-intercept: (\ Y\g Asymptot Increasing: Decreasing: t\\a. As x > -oo, the graph approaches -?!,' For 7 = log 3 (x - 2), fincrfhe following. ^-intercept: ^ 3^0) y-intertept: D:_ ix R:_ Asymptotes):' Increasing: _ Decreasing: As x -» -oo, the graph approaches

17 5. Using transformations, graph y = Iog3 k As «s 0, i 7. Using transformations, graph y = -Iog3 x. t- «5P^ " > -^ -- *.v-interceptjf Tjb', y-intercep D: R: lu. Cj^Cfci As ymptote(s):* "X "*-O In> ;reasing: A\^. De creasing -«< x ^ O As x» co, the graph approaches As A:» -co, the graph approaches As x -> 0, the graph approaches _ f t: f icirvj jr-intercept: D: -intercept: f\qa> Asymptote(s): X^O Increasing: Decreasing: Q As x» oo, the graph approaches ~ As x -> -oo, the graph approaches As x > 0, the graph approaches <=**=* 6. Using transformations, graph y = 2 Iog3 x. 8. Using transformations, graph y- K>g3(X + J)-KZ. t, Uf 2 -, r==«!^ -? ^-intercept: HjO/ y-intercept:_ D: X X I X > frl R: Asymptotei Increasing: _ Decreasing: _ As x» oo, the graph approaches _ As x > -oo, the graph approaches 0 " -=> -** fai >) xj ^-intt:rcept: y-intercept: *^t ^ D: X ' '3J R: V I * trtt. Asympto're( s): - 5 Increasing: Decreasing: r -3^>c<*o Kit As x > -co, the graph approaches -7-3 ^?