MATH 1051 Lecture April 7,2017 29 Announcements 5. I & 5.2 Today Read Section 5.3 for next time 5.1 Composite Functions Functions Some Define f (× ) ×2+ = ) flymeans functions ¥+1 , g - Review e.g. ( (4) 2+24-+1 x ) 3fF2 = ( replace every but don't We can do math to get a value f ( 4) " x " with " change anything = 16 + 4 " else ) 's +1=17.5 goes lgtftfniinminanh.tmpowariii.in Functions of Instead plug In f (× ) × We g( xth We ) = did number a like , 2+2×+1 )=tH't ftx in expression an = plugging IT did g this to - ( x ( replace +1 to 3(x+h)-2f this Review - test find the 4 X = even every We , Can xth or zxfz " every for ( replace ) like , / " difference x " odd x " ' with ' × - " ) . with " Xth quotient " ) Composition of f(×)= So , × what f(g( A) 2+2×+1 woud = ( fcgc xD )2t g mean # ? + 1 Functions ( x ) = zxfz Composition of So , f(×)= ×2+ what woud fcgc A) Say Called Can " f = 3€ ¥+1 fcgc xD g ( x ) = ? 3*22+(3×2.7) ( of + g of the write Mean Functions × " ition fcgc xD 1 OR # of fog meaning Jame fandg ( ×) fog ( Find fk FI )=2X fog Q (x) find = ( × ) = , (×) f( gcx) ) = 2 ( ×+÷ ) goh ( x ) A) 2×3.1×27 B) 21¥13 ( EI Find hogk ) hlgcx ) )=2 Q Find ) X2 g fog find ×÷ x # hof ' hk )= = 2×3 . ,2€ # )2C)ak+D3#xn2 D) ? Hx÷i)±24¥ ) x¥p=¥T5x¥y - ( ×) A) 2×3-2×2 B) 4×3-2×2 C) 16×3-4×2 D) ? fog Find fog is a function too , , so we # ) can find fog fog C- 2) f ( ×) fog (8) = f(g( gof 732 Q Find A) 3×2 = g 8) ) ( If D= =3 ( FF 2=3 ) ( 2) 2=3.4--12 (3) B) 33/32 C) 3 D) ? (2) . or fog (4) or f 09 #) Find find ¥ goft⇒ jkeo.lt • ( 0,4 ) f Looking for (× ) g(ft5D ' "ta¥e±⇒ Then g( x ) find g(o)=4 goftst glo ) . 4 . of Composite Domain h (x) Vx = g ( ht D Find EI glhk ) ) -1 not is ( EI g (x) , the in because Find -1 g( g ( hl 4) ) 4 is b( ecause EI 2 ( II event in hl 4) =2 find g( in the though h not is = not (a) ) , in # the Of = NOT - ¥ 2 is not in of in ~~ glh the - Uh oh ! of h (xD to = of ¥ = domain domain ¥2 1) g(h(xD ! the = and domain 2 is glhl . domain ( 4) ) h ¥2 = # = function uh oh ! glh ( xD ! the 1.7 of domain g(xD hooray ! ( x )) domain of g ( x ) ! Wacky! ) of Composite Domain E± f ( x ) Find ( domain ¥32 = h) = of domain the of h (x) Foh +1>-0 2-1 × × ( domain of f) : hlx ) -31=0 hk)±3 Fit X ¥3 + X I 1 t 9 8 {111×2-1,1*8} At ( x ) . function f Find EI Find A) fk B) f functions fk ) = ; ( KI )=I x ) & g gfx ) ; g(x)= and g such (x ) xt = KI that fog (× ) = ¥1 f (g ( x ) ) When EI f ( x) X gof x 3f2X ) ) = (x ) So , fog Show that fog ( g ( ×) 2X =3 = -2 fog ( x ) ( x) = - = goof = 6X (3/1)=-6/1 = g of (×) - . ( x) . = gcfcx)) 5.2 One - to - One , Inverse Functions One functions ALL . : to One - output One Functions per input / Not Function y=rx if X=4 ,y=2 One Fro - to - other if functions one : y=20R , -2 ( oneinputperoutputlltgirnizofetsaf Not EYE 's / '¥i¥ et # ×=4 test function a y=±rx - One-to-One line Vertical two one inputs to - . one - one output Functions Inverse f ( × ) =×3 / Graphs Fx an f. g(×)=F× Reflection : Algebra Called ×3 between Relationship : # inverse fogk ) = x = gofcx) and functions If the across , = line ( X ftp.X write f(x)=×3 - f- " , y=x Identity . ' ( x) . ( × ) =3f× function ) Must have To one EI - to an one - One Beto function inverse ( or "l - l ' ' NOT 1- 1 f ( x ) One must be ) f- ( x ) ? ? ' flxkxzqg - . Refle Ct across g Not a function y=x Verify functions Inverse Operations E±: fory Solve Capture do the " or Inverses Function Function " Opposite of that that adds : " undo FK ) subtracts 3 is subtracting } =x+3 3 ! ×) " solving equations when 6-3=9 = adding 3 Can 6 y+3= " we way " opposite y The the g( = X -3 . Verify EI fog Verify (x) ( x 3) = go F ( x ) that - = (g f- ( × ) × = ( x+3 ) -3 = ' gk ) +3 Inverses = So × g EI Let gk ) = tx glhlxll hlg (A) and hlx )=4x 'T ( 4x ) = = 4 ( tax) x =x inverse function of f) ' = f- ( × ) Verify , . = (× ) the is . that g- ' k) =hk ) . Find To find f- ( ×) ' the Replace 4 X 3 =3y 4 - X - 4 34 Find . +4 X=3y 2 the f- ( x ) ' =y = ¥ in y variables × &y y Replace y 3×+4 y= 1 for Solve 3×+4 = with Interchange 3 fcx ) fk ) equation 1 , 2 EI Inverse ' f- ( × ) with ' f- ( × ) . in the equation Find Q flx ) @ 2 3 4 = ¥4 the , Inverse Find f- ' (x) . Find E± The of graph f- ' ( ×) of the flx ) Inverse is shown . Draw . y=fK ) the graph Restrict about What inverse an If find of 1 2 3 4 to XZO ×=y2 the fx=y ' (x) Nol 1- I doesn't so , have . gk ) y=x2 g- ? gcx )=×2 function EI f(x)=×2 restrict we Domain the now , function inverse 1- a I / XZO , have we xzo , yzo , ( =V× no ± ( fk ) because 70 it but restricted we we because don't rx need is to always to write 201 yzd

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