Edexcel A Level Maths Year 2 Chapter 2 Functions

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the graphs ) Solve the inequality - 13x+41 42x-9 and 3x+4=2x-9 x=-13 -B3x+4)=2x-9 -3x-4=2x-9 SS x=1 12: Solve the inequality 12x+9/214-x -12x+91=14-X 2x +9=14-X -2x-9 = 14-x 3x = 5 2=¾/3 225/3 (x:x>13 (x:x²-133✓ of 2y = - Bx+4) and y= 2x-9 x=-23 x>-23 Ex:234x25/33 13: The equation 16-x) = 1/2x tk has exactly one solution a) Find the value of K. the equations must intersect at (6.0) for their to be excectly one solution y= ¼/22+k 8=1/2(6) HK K=-3 b) State the solution to the equation. x=6 14/09/21 Examples: Mapping 1) The function fod is clefined by £: x + (5 273,1 x 7/1 x41 of fles. a) sketch y=f(x), and state the b) Solve fu=19 1²+3=19 x² = 16 x=4 1. The function & is defined by fix →→e* +2 State the range. df. 16072 -4x+1=0 (x-2)²-3-0 5-8)²-3=6 2: The function g is defined by g: x →→x² - 4x +¹₁ x ER, 0≤x≤5 the range 29. turning point ad (2-3) -3496046 a³=46 a = 3.58 3< لحا؟ 2 B 8: The function p is defined by PGJ (234, 55 x 54 Sex, @) Sketch y=pla a) Sketch y=h(J b) Write down the range of f(x) 2 ≤h (w) ≤ 27 5-2x=19 -2x=14 2=-7 b) Find the values of a, to ? decimal places, such that pla) - SO e = 50 a³+4= 50 -a = ln so a=-3.91 range m= 27-2 = ¾/2+(.. 6-64 9: The function h has domain -10≤x≤6, and is linear from (-10,14) to (4,2) and from 64,2) to (6, 27). c) Find the values of a such that h(a) = 12 equation of 12 = moetc a=0 equation of 11= moete M = 14-2=-6/7-2 (-10)-(-4) 2=-6/7(-4)+( (=-40-6 XER 12=-677a-486 a=473-9 16 -24 L₂a FIR 10 22 (-10)-69 Example: 1) 2 C The functions fand g are defined by. fix12x-81 gixxt a) Find fg (3) R$20=1266244-81 a) Find fg(1) fg= (3-42) +3 fg(4)=13-4(0/+3 =1+3=4 = 6x6-71 fg(3)= (3-71 = 4 b) Solve fg(x)=x 1x-71=x Is The functions & and I are defined by fixtolx +3.x EB 9:x² 3-4x, x ER 8xx-1)=0 x=0 and x = ¹/2 Composite Functions 6) Solve the equation gg6d) + [g(e)] ²=0 (3-4x)² = B-12+8x] + (9-24H) -0 18x²-161-0 2. The functions fand g The functions fand = X-2 y=x Ag(z) g a) Find fg (se), giving your answer in its simplest form. Rg6d=1+² = x+2 3+4 a) Find an expression for bg (c). fg(x)=32²-2 b) solve fgle = gf6d₂ 3x²-2= (3x-2)} 3x²-2 =9x²-12x+4 6x²-12x 16-0 - 1x-71=x -2+7=X 2x=7 are defined by fixe ²+² +2.x EB gxx xx0 are defined by x=7/2 4. The functions p and 9 are defined by p60) = 1/2-²2, x ER₁ x + 2 9l = 3x+4₁ x ER a Find an expression for a plc) in the form orth ap6d=31)+4 (x td = 3+46-2 x-2 = 3+4x-8 = 4x -5 X-2 x-2 f(₂)= 3x - 2₁ x ER g(x) = x²₁ x ER x²-2x+1=0 (x-1)²-0 x=1 a = 4 b=-S C=1 d=-2 4:6) Solve qp60 = 16 4x-5 = 16 x-2 4x-5-16x32 12x=27 x=9/4 5: The functions I and I are defined by fixt 19-4x1 9:xt 32-2 a Find fg(6) 89(2)= [9- 4(32-²)] =19-6x +41 = 15-6x1 fg6) = 15-6(61 = 31 6) Solve fg (x) = x 15-6x1=x -15-6x=x 6x-5=x 5x=5 x=1 X 6: Given f(₂)= 1, x=-1 xts a) Prove that f'(x)=x+1² 4 x+2 8² (2)=(1) x+1/+1 = 1= x+² = xtl X+1 x+2 b) Find an expression for £3(x) (1/x+1) 11 (1/x+D+2) = 1+2 = 2x+3 x+1 2+1 = (x+²)(x+1)=x+2 (1)(2x43) 2x13 7: The functions s and + are defined by slex)=2*, x ER th-x+3, XER a Find an expression for st(x) st (x) = sin = 2²x2²³-8×2² b) Find an expression for ts(x) ts(x) = (2x)+31 Sdue stG)=ts (₂), leaving your answer in the form Ing 2²15 = 2² +3 2x+3 Inb -2x = 3 22x2³-2²-3 288-27-3 2* (8-1)=3 2x=3 7 xtin2 = In 2 7 x= In (3/7) In 2 8 Gven f(x) = e ³x and gly = alinx, find in its simple, wo es form: 5(linx) a gf(x) = 201iax - 20x =e 9. The Unctions pand q are defined by pix1 • In(x+3), x €, x>-3 q: xe"-1, x ER a) Find apo marty ap) = e² (state its range = (x+3) ³-1₁ 9000>1 (6) Find the wolve o ap (7) qp(7)=(713)31 =999 c) Solve ap(x)=124 (x+3)3-1-124 (x+3)² =125 10. The function t is defined by t:x15-2x the 5-25-2x)-(5-2²=0 5-10 tu-(4x²-20x²+25)=0 -4x² +24x-30-0 2(2) = 12 + 144-120 U 2x²4-12x+15-0 x=-(12)=√(12) ²-4(2X15) x+3=5 -8≤ god ≤ 12 x=2² b) Find gg(0) g(x) = mxtc (5-12)+12 (140J =-11/2x+12 J Find gh(7) 11: The function I has domain -5 ≤ x ≤ 14 and is linear from F.S. -8) to (0, 12) and from (0, 1) to (14, 3). a) Write down the range d g.. g(0)=12 The Runction his defined by h:x+ 2x-5 10-X ghl₂)=-1/2 (2x-5)+12 = 5-2x +12 20-2x x= 3± √24 4 gh (7)=5-2(7) +12 20-21) = -√2+12 = 13.5 10.5 = 3+√√√ 2 99(0) = -4/2/1/2x12 +12) +12 = -4/2 (18) +12 = 60 fg(x)=x20 ● 24/09/1 Inwese Functions • domain of Hx) is the range of f(x) and vice versa Examples: 1) If f(x) = 3-4x, find f (₂) у=3-их 'f- 4(x) = 3 = x 4x=3-4 x=3-5 2) If f(x) = x+²₁ x #4/2, determine f-1(₂) 2x-1 Lety=x+² 2x-1 2xy-y=x+2 2x²y=x=yt? x (ly-1)=yt? I: If glad is defined as g(x)=√x-L, Ex E18, x7/23 a Find the range of glod g(2)=0 g(x07/0 b) (alculate g(x) lety=x²-2 y² = x-2 x=y²+² g-1(x) = x² +²² d) Sketch the graphs d both functions x = yt? 25-1 1-40=x+² xxx-1 d) State the domain and range of g range: g: ¹x)7/2 domain: 27/0 g16) glx) graphs reflect at y=x 2: The Unction fis defined by f(x)=2c² 3₁ x ER³₁ x 30 a Find f (x) Lety=x²-3 y+ 3 = x² f- + (x)=√x1³3 x = √y+3 b) sketch y = f(x) and y=f-4(₂) and state the comcil of f-160) f(0) = -3 domain of 1-1/2) = x ²1-3 y=f-8 4 = POS 3: The Unction fis defined by fixe² +²₁ x EM (a) hind ft. the inverse function of f stating its domain. lety=e²+ + ² y-2² = ex x = In (y-2) b) On the same axes sketch the curves with equations y=f(x) and y=f=1(e), giving the coordinates of all the points where the curved cross the axes. 205: The function food is defined by th) = x²6x+ 5 x €R₁x75 Find & H. lety=x²-6x +S + 4(x)=√6(+4) +3 = x уже = (х-3) г Sytu = x-3 x = √y+4 +3 6: The Runction m() is defined by m(x)=x² +4x+9₁ x ER, x> a, for some constanta. a state the least value of a for which im F(x) exists.. |m60) = (x+y)² - 4+g minimum point at (2,5) helle x>-2 = (x+2)² +5 a=-2 6) Determine the equation of m² + (x).. let y = m(x) y = (x+2²+5 y-s= (x+2) xy-5)-2=x Ostate the domain of m² + (x) range of m(x): m(-2)= (-2)² +4(-2) +9 =S domain of m² + (x): x>5 XEHR 7: The function W(x) is defined by NGJ) = 2x+1, {x EB,x± 2). a what happens to the function as x approaches 2? as x approaches 2, h(0) hence how 6 m²(x)=√x-5-2 b) Find n-¹(3) (n +(x): lety=h(x) Ў= 2х1 X-2 xy-2y = 2x11 2x-xy=24t1 (a) Find nm() x = lyti 2-9 c) Find 2), stating clearly its domain. (n-1(2) = 2x+1, XEDB, X 2-x h-+(x)=2x+1 2-X ѝ4(3) = 2(3)+1 2-(3) nm(x) = (2x+3)=3 = 10 # ニーモー 8: The functions in and I are defined by mix+ 2x+3₁ x ER nix (x-3/1/2, XER = b) What can you say about the functions mandn? mn(x) = {(x=²) + 3 = x The functions mix) and n(x) are the inverse of each other as ma(x) = x=nm(₂) O ● 9: The functions s and it are defined by s(₂) = 2/₁2 #1 16x)= (3-2/xx10 Show that the functions are inverses of each other. sta) = 3 = 3÷ (3-2 +1) 3-21 11 = 3- (3-xxx) =3=3/4 =x 4560) = 3-1 2-2) = 3 - ( 23 ) = (27) x+y 3 = 2 as +5(x)= x= st(₂), the functions food and slod) are inverses of each other 10. The function is defined by f(x) = 22²=3, {XEB, X403 Determine: =2x-3 = 3x+33x xt1 x+1 그렇 = 3x 23 all (x) clearly staling its domain. let y = f6cr yt3 = 2x² 43=x² X= 2²²60)² = √23₁ x ER₁ x 4-3 Sytz at f(0)=-3 f(x)2-3 b) The values of a for which f(a) = f-1(a) 29²-3-a 16²-3-6-0 (7) (943) (20-3)=0 à-=-1a3/2 as we can see that the point of intersection is negative hence a=-1 24/09/21 Examples: 1) 2) b) y = f((zl) Thething y=14601 J y = f(bal) This is a sketch of a) Sketch y=1f60)| b) y = P(x) b) Sketch y = f(b) 1. Figure 1 shows part of the core with equation y=f(x) The curve passes through the points PC1. 5.0) and B(0,5) as shown. On separate diagrams, sketch the curve with equation a g=18601 Jos ELSO التاو = و له b) y = q (lx) Sketch for -2Fc≤ x ≤ 212: ay = | singl b)sin/col 2 E 5: The diagram shows the graph of y=plz) with Sponts labelled. sketch each of the following graphs, labelling the points corresponding to A, B, CD and E, and any points of intersection with the coordinate axes. @ y = 1pW) ME y = f(x) where f(x)=(x-3)(x+1) P 6: The diagramshows the graph of y = y(x) with 7 points labelled. sketch each of the following graphs, labelling the points corresponding to A, B, C, Dand E, and any points d intersection with the coordinate axes. 49 JUL 2 TO 14/1/2000 4 7 AUDY Lata 40 • ● 24/09/21 Combining Transformations transform x-coordinates befory-coordinates in compared transformations? Example: 1) Here is a graph of y=f(x). • Sketch the graph of y= 2 + (x+²) [yild) 15 1: Figure 1 shows pult of the graph at y = f(x)₁ x EPB. The graph consists of this line scyments that meet at the point R. (4,-3), as shown in figured. sketch, on separate diagrams, the graphs of: [@]y= 2+(2+4)" 16₁-6) b) y=18(2)] RGD 2 F 4. The functiong is defined by gux+62-2²-9₁xER a) Drow a stretch of the graph of y= g(x), labelling the furning points and the x-and y-intercepts (ii) g (2x) (i) 19(2) minimumpoint at £2,₁-9) •y intercept: y = (0-21²-9 (0₁-5) = 4-9 -by = -S b Write down the coordinates of the turning point when the curve is transformed as follows: (i) 29(x-4) furning point at (2-9) new turning point at 1/2+4), (-9x2) = (6₁-18) turning point at 12-9) = (1,-9) turning point at 42, 1-9x-1) = (2,9) (c) Sketch the curve with equation y = (g bd). On your sketch show the coordinates of all turning points and all x- and y-intercepts. tokoll tumly pint = (2, 9) x-intercepts=(-1.0), (5,0) y-intercept(0.5) At 6: The diagram shows the curve with equation y=f(x). Find the coordinates of the paints to which O, A and B are transfamed on the curve with equation y=f(1/3x+2) O 49 (y₁h (₂) A: T B: 0-2=-2 -2x3=-6 (-6,0) 4-2=2 2x3=6 (6,-2) 7-2-5 Sx3=15 x-intercepts: 0=(x-32-9 9-x-2² -3=26-2 3=x-2 x=-1 x=5 (15,0) 18/09/24 Phetching Hoctalus Starter 1) The following table summarises the times, + minutes to the nearest minte, recorded for a group of students to complete an exam. 2 G Time sites) + 11-20 21-25 26-30 31-35 36-45 46-60 62 88 16 13 11 10 [You may use Σff² = 134281.25] a Estimate the mean and standard deviation of these data. 1 Given that f(x) = 2-√x+11. a) Find the coordinates of the paints P. Q and R. R. 0=2-12+11 Q: - HI 9= 2-10+11 = 2-1 2=x+1 x=1 mean= Ex = (53x10) + (40.5+1)+(33x13)+28 x16)+(23x88)+ £15.5x62) 62 + 8 8 + 16 +1 3+ 11 + 10) = 4837.5 = 24.2. 200 b) Use linear interpolation to estimate the value of the median.. dian= Ex² (5) ² = 134, 281 25 - (24.2) ² n 200 = 9.29 b) Shade the K(10) b Save $(0) = 1/2x 1/2 x==2-bct 1) 112x=2-x-1 3/2x=1 2-4/3 region Q (0,1) 2. Given that p(x) = 2bct4l-5₁ x ER, a) Sketch the graph of y=plz) (280) (320) (-4,-5) PEWN 1/2x=2+x+1 -1/2x=3 x=-6 10= 26c+41-5 5/2=x+4 20=-3/2 y = 20+41-5 9=3 that satisfies y p(x) 3. Given that g(x)=-3 bxl+6, XER, (₂) Sketch the graph of y=9G) 4x+25=-1/2xt 1 9/2x = -24 2-46/30 6=-36x146 2 = 12c (-20) (6) x= 2₁x=-2 5) Shade the region of the graph that satisfies y ≤ p(x). Solve the equation f(x) = -4/2x+1 4/x+61+1=-1/2x+1 5/2=-2-4 x=-13/2 4. The dunction f is clofined as +x+ 4/x+6/ +1, x ER, a) Sketch the graph. of y= :f(x). (b) Stake the range of the function.. +601 NOTE at y coordinate of P = 2 2 -1x+11 = 2 *x+1=0 x=-1 P (1,2) - 4x-23= 1/2x+1 - 7/2x=24 X=-48/7 1Q ● ● 5. Given that g(x) = -5/2 17-21 +7, XE8₁ a) sketch the greeph of y= glu (b) State the range of the function. PGU≤7 O Save the equation g(x)=x+1 -5/21x-21 +7=x+₁ -5/2x+12=xt1 axy x=14/2 6: The functions m and in are 562x+2=x+1 3/2x=-1 x=-413 • defined by -415 8: The function h is defined by WCx) = 2/3 |x-11-7, XER a) State the range of h. h67-7 M The equation mcd=16d has no real roots. And the range of possible values for the constant minimum point of A(₂0) = 14, 6) K. m(x) must be less than this to not intersect M (W)=&th -8th 26 KLJU Ws m(x) = -2x tk, x EffP n(x)=3(x-W 16. x ETTB where Kis a constant 7. The functions s and I are defined cos(x)=-10-x, x€ IR where bis a constant t(e)- 2ktbl-8, XEB The equation s(c) = Hoc) has excactly one real root. Find the value db. one root means dunctions intersect at trorning point of 160).. turning point at flo, -8) s60-10-x -8=-10-(-b) b=2 y (6) Give a reason why hot dees not exist h() is many-to-one (when y=0, there are two value) hence n`t does not exist as hite) would not be a function. 1) solve the inequality h(x/²-6 2/3/x-11-72-6 x-143/2 GC20 XL 5/2 -20163/2 X742 d State the range of values of K for which the equation hod) = 2/3 xtk has no solutions. 4/3x+K must be less than minimal point of h(x) for there to be no solutions. 2/3x+12-7 42 KL-13/8 9 The diagram shows a sketch of part of the graph y=h(x), where h(x)= a- 21x+31, x ERR. -axis at (0.4) The graph intercepts the a) Find the value of a. h(0)=4 a-2k31=4 a-6=4 a = 10 b) Find the coordinates of P and Q n(x) = 10-2/x431. Q: 0= 10-2/x431 0210-2x-6 2x=4 x=2 b) y-coordinate of P= 10 10=10-2/x+3) 23-0 X=-3 Solven(x) = 1/3 26+6 10-2bx+31=1/3x+6 10-2x-6=1/3x+6 -7/3x-2 x--6/7V 10 The diagram shows a sketch of part of the graph y = m(2), where mb= -4/x+3/+7, XER a state the range of m... mG067 P(-3,10) b) Solve the equation mGc= 3/5x+2. 3/5 XH2-4b4347 3/5x+2=-4x-12+7 135x - -3 x=-15/23 10+2x+6=1/3x+6 5/3x = -10 x 4x4-2612 x=-5 Given that muc)-1, where K is a constant, has two distinct roots. o state the set of possible values fork. 1-4/2+3/+7=K -4x-12+7=K -4x-S=K T wxt121=k их +19-к for there to be two foots, k must be <m7 4x+19 47 -4x-5407 XL-4372-3 K is simply (7 3/5x+2 = 4x+12+7 15x-17 -4x412 4x7-12 x> 11/23 1 ● 21/09/21 1: Given fl=e²₁xE18 Frundions & graphs 960 = 3 linx, x20, XER Find an expression for glow simplifying your answer. gf(x) = 31in(ezy = 3x 2: The function f is clefined by f:x+3x-5/x+1, XER, x=-1 a show that ff(e) = 2+q x EP, x ±± 1 where a is an integer to be found. ff6d= x-1 3 [ ²³ (37=-5) = 5] = [ (3x = 5) ++] = (3x-15-5x-5) = (3x-5+2+1) - = 4x-20 x xt²t x+1 4x-4 b) Find f(x). let y= 3x-s x+1 xyty= 3x-5 9 + 3 =-=-32-gy y+ 5 x(3 8 (5)=(5)²-3(5) =25-15 = = 10 range of g: -9/4₂ ≤ glx) ≤ 10 42-20 4x-4 46(-5) = x-5 4 (x-1) x-1 3: The function g is defined by gixx²-3x, XER, OLX45 Find the range of g g(0)=0 α=-Su x = yts 3-4 1-²(x) = xts, X²EB X 3 3-X g6x)=(xx-31)2-(3/2)² =(x-3/2)²-9/4 9min=-9/4 24/09/21 Imerse Functions. 14. The functions of and I are definal by fixtex-s₁ x EIB 9:x In (x-4),x24 a) State the range off. S6> Su b) Find & the inverse dunction of f, stating its domain. f++ (0) = Inbets) demain: x>-5 let y = f(a) y = e²-s y+²5=e² x = (nyts) On the same axes, sketch, the curves with equation y = f(x) and y=1-160), giving the coordinates of all the points where the curves cross the axes d) Find g-t, the inverse function of g, stating its dermain... lety = gla) 9²¹ (₂)= ex +4₁ XER ((n (x-4) yos ay 2= 1² +4 e) Solve the equation g=+(x) = 11, giving your answer to 2 decimal places. @²²+4=11 ex=7 x = ln7 2195 + 12. The function & is defined by fixt 3G+2) x ²4x-16 a Show that fixt 1 = f(x) = 3kct2) (27512-4) 2+51x34 36(+2)-2(x+5) (x+5)(x-4) b) Find the range of f 11W= 1 = ¹/g 4+5 x+5 xy + Sy=1 y = -5 2 (x-(1) xy = 1 - sy - 16x1/9 c) Find £²6. State the domain of this inverse function. | let y = f(x) 12274 x-u 3x+6=2x-10 - (x+5)(x-4) 8-1 (20) = 1-5/ x domain: x1/9 (1+3)(x-41) = 481024 x45