`Worksheet for:function notation difference quotient naming the domain of a functionFUNCTION NOTATION EXAMPLE: For the function G defined by G(x) = 2x2 - 5x, G(input) = 2(input)2 - 5(input) Evaluate: a) G(6) b) G(-1) c) G(a) d) G(b) e) G(h) f) G(x+h) Answers: a) 42 = 2( = 2( = 2( =2( =2( =2( b) 7 #2 c &amp;B )2 - 5( )2 - 5( )2 - 5( )2 - 5( )2 - 5( )2 - 5( ) ) ) ) ) )EXAMPLE: For the function defined by f(x) = 4x + 1, Write the corresponding ordered pair. f(input) = 4(input) + 1 a) f(3) = 4( ) + 1 = b) f(-1) =4( ) + 1 = c) f(a) = 4( ) + 1 = d) f(b) = 4( ) + 1 = e) f(h) = 4( ) + 1 = f) f(x+h) =4( Answers: )+1= +Ñ Ð \$ ß &quot; \$ Ñ /Ñ Ð 2ß % 2 b &quot; Ñ Ê( , ) Ê( , ) Ê( , ) Ê( , ) Ê( , ) Ê( , ) ,Ñ Ð c &quot; ß c \$ Ñ 0 ÑÐ B b - Ñ Ð +ß % + b 2ß % B b % 2 b &quot; Ñ &quot; Ñ .Ñ Ð ,ß % , b &quot; Ñ  0 Ñ #Bb% B 2 bc &amp;2  c) #+ c &amp;+.Ñ #, c &amp;,/Ñ #2 c &amp;2 ExercisesGiven the functions:f(x) = kx - 2k, h(x) = 2x4 - 5x2 + 3,g(x) = 3x - 5, k(x) = 4xfind each of the following: 1) f(5) 7) k(0) 12) h(a) answers: 2) f(-2) 8) k(3) 3) f(2) 9) k(-3) 4) g(1) 10) h(1) 5) g(0) 6) h(0)11) g(a) (replace x with a) 14) f(x+2) 4) -2 11) 3a - 5 15) 4-x or ( 1 )x 4 15) k(-x) 5) -5 6) 3 7) 113) g(x-4) (replace x with x-4) 1) 3 8) 64 13) 3x - 17 2) 4 9)1 643) 0 14) kxk 10) 012) 2a4 -5a2 + 3REVIEW OF SLOPE Recall the slope formula for an line: In the linear function f(x) above This can be written using function notation:slope = m = m=13 - (-3) 3 - (-1)y1 - y2 x1 - x2 16 4==4f(3) - f(-1) 3 - (-1)For the two points (a,f(a)) (b,f(b))f(a) the slope = m = f(b)--(a) bFor the two points (3,f(3)) (h,f(h))f(3) the slope = m = f(h)--(3) hFor the two points (x, f(x)) (x+h, f(x+h)),the slope = m =f(x+h) - f(x) (x+h) - (x)=f(x+h) - f(x) hThis last expression for slope is called the DIFFERENCE QUOTIENT, h Á 0. It is used often in calculus to find the slope between two points of a function.DIFFERENCE QUOTIENTf(x+h) - f(x) hh Á 0Find the difference quotient for the function f(x) = 4x + 1: Solution: Break this difference quotient into pieces. f(x+h) = 4(x+h) + 1 = 4x + 4h + 1 Now subtract f(x) from this result.f(x+h) - f(x) = (4x + 4h + 1) - (4x + 1) = 4h This is the resulting numerator. Put the pieces together.oe %Find the difference quotient for the function 0 Ð B Ñ oe c #B This is the resulting numerator. Put the pieces together.oe c#¨ ¨  oe¨ 4  98¤   £51¨   8¤    ¦ 4 ¦ ¨ 4 ¦ 05§¤   7&amp;650©§¢    ¢  4 ¦  ¨ ¦ ¤oeoe¨ ¨ 0oe¨ #  ¤  # ¦ ¨ ¦ ¤ &amp;1(32\$10§(b \$¨  # ¤ ¢  # ¦  ¨ ¤¢ )\$¦ ('&amp;%\$&quot;!©¦ ¥oeoe¨  ¤¢ ¡   ¨ ¦ ¤¢ ©§¥£¡ ¨  ¤¢ ¡   ¨ ¦ ¤¢ ©§¥£¡Find the difference quotient for the function f(x) = 2x2 - x + 1f(x+h) - f(x) hè ë ë ë ë ë ë é ë ë ë ë ë ë ê [2(x+h)2 - (x+h) + 1] f(x+h)h Á 0f(x) è ë ë é ë ë ê (2x2 - x + 1) =subtract all of the quantity f(x) = [2(x2 + 2xh + h2 ) - x - h + 1] - 2x2 + x - 1 =2x2 + 4xh + 2h2 - x - h + 1 - 2x2 + x - 1 = 4xh + 2h2 - h combine like termsNow divide by the denominator hf(x+h) - f(x) h=4xh + 2h2 - h h= oe Put the pieces together. 0 Ð B Ñ oe % B b &quot;h(4x + 2h - 1) hfactor divide out the h4x + 2h - 1oe% BThis represents the slope between any two points of the function f(x).F F QP h Gpy F U E F CoeF b P C E h C U P b E F P C Gswv2tuXptxfP h £Gpv2U F U E F C y E h CoeF b P C E h C U P b E F P C P H h F E F C U Yxwvtugptsr1qapi2GE h RWU CAb #2 c &quot;F H b E C 2g`fP h 2eYdXWIa`DYXIG\$RWVT C UA P c b E H F E CA P hH F E CA UoeF H CA @ P H F E CA [email protected]oeEXERCISES For the functions f(x) = 2x - 3, g(x) = x2 + 4x - 3, find: 16) f(x+4) 20)f(x+h) - f(x) h17) f(x+4) - f(x) 21)g(x+h) - g(x) h18)f(b) - f(a) b-af(x+4) - f(x) 4 g(b) - g(a) b-a19) f(x+h) - f(x)22)23)answers:16) 2x + 5 21) 2x + h+ 417) 8 22) 218) 219) 2h 23) b + a + 420) 2EXERCISES Evaluate the difference quotient for each function. Simplify. 24) f(x) = 1 - 3x 27) f(x) = 2x - 7 answers: 24) -3 25) 6x + 3h - 2 26) x(x+h)-125) f(x) = 3x2 - 2x26) f(x) =1 x27) 2NAMING THE DOMAIN OF A FUNCTION Find the domain of each function: 28) f(x) = 3x2 - 2 31) f(x) = È x + 1 29) f(x) = 32) f(x) = È x2 - x - 2x x2 + 130) f(x) =x-2 33) f(x) = É x - 4x x2 - 1answers:29) All real numbers30) All real numbers except ,, 1 31) x   32) x   33) x Y -1 In interval notation: -1 Ò c&quot; ß _ Ñ Ð c_ Ð c_ ß c&quot; Ó r ß #Ó r Ò #ß _ Ñ Ñ2 or x Y 2 or x &gt; 434) -2 &lt; x &lt; 234) f(x) =x 4 - x228) All real numbersIn interval notation: Ð c _ ß _ Ñß _ÑIn interval notation: Ð c _In interval notation: In interval notation: In interval notation:Ð % ß _Ð c #ß #Ñ`

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