Functions — Paper 4 · IGCSE Mathematics 0580

(a) Find f(1). … [1] (b) Solve f (x) = 3 . x = … [1] (c) The equation f ( x) = k has only one solution for - 2.5 G x G 2 . Write down the range of values of k for which this is possible. … [2] (d) By drawing a suitable straight line, solve the equation f(x) = x – 5. x = … or x = … or x = … [3] (e) Draw a tangent to the graph of y = f (x ) at the point where x = 1. Use your tangent to estimate the gradient of y = f (x) when x = 1. … [3]

10 marks

Mark scheme: 4(a) –1.6 to − 1.4 1 4(b) –0.5 1 4(c) k > –4 2 B1 for identifying the –4 or for horizontal line drawn y = –4 4(d) y = x – 5 ruled 3 B2 for correct line and 2 correct values or and no line and 3 correct values –2.3 to –2.1 or B1 for no line and 2 correct values –1.2 to –1.1 or B1 for correct line 1.3 to 1.4 4(e) Tangent ruled at x = 1 B1 No daylight at point of contact. Consider point of contact as midpoint between two vertices of daylight, the midpoint must be between x = 0.8 and 1.2 –6 to –4 2 Dep on B1 or close attempt at tangent at x = 1 M1 for rise/run for their tangent at x = 1

This question in 0580/41 May/June 2017

10 f (x) = 3x - 2 g (x) = x2 h (x) = 3x (a) Find f (-3) . … [1] (b) Find the value of x when f (x) = 19 . x = … [2] (c) Find fh(2). … [2] (d) Find gf ( x) + f ( x) + x . Give your answer in its simplest form. … [3] (e) Find f -1 (x) . f -1 (x) = … [2]

10 marks

Mark scheme: 10(a) –11 1 10(b) 7 2 M1 for 3x – 2 = 19 or better 10(c) 25 2 M1 for 3 × 3 x − 2 oe 10(d) 9 x 2 − 8 x + 2 final answer 3 M1 for (3 x − 2) 2 + 3 x − 2 + x oe B1 for  ( 3 x − 2 ) 2 =  9 x 2 − 6 x − 6 x + 4 oe   10(e) x + 2 2 y 2 oe final answer M1 for x = 3y – 2 or y + 2 = 3x or = x − or better 3 3 3

This question in 0580/42 May/June 2017

7 f(x) = 3 – 2x g(x) = , x ≠ 0 h(x) = 4x x (a) Find f(5). … [1] (b) Find gh(3). … [2] (c) Find f –1(x). f –1(x) = … [2] 64 (d) Show that hf(x) = x . 16 [3] (e) Find the value of x when h(x) = g(0.5). x = … [2]

10 marks

Mark scheme: 7(a) –7 1 7(b) 4 2 4 or better M1 for g(43) soi or or better 64 4x 7(c) 3 − x 2 y 3 oe final answer M1 for x = 3 – 2y or 2x = 3 – y or = – x 2 2 2 y − 3 or oe as final answer −2 7(d) 43 – 2x M1 Correctly interprets the indices M1 Dep on previous M1 1 43 e.g. 43 × 4–2x or 43 × or 4 2 x 4 2 x 64 A1 nfww Correct completion with no errors 16x 7(e) 1.5 2 B1 for 4x = 8 or better

This question in 0580/41 Oct/Nov 2017

9 f ()x = 1 - 2x g ()x = x + 4 h ()x = x 2 + 1 (a) Find (f-1) . … [1] (b) Solve the equation. 2f ( x) = g ( x) x = … [2] (c) Find fg ()x . Give your answer in its simplest form. … [2] (d) Find hh(2). … [2] (e) Find f - 1 ()x . f - 1 ()x = … [2] (f) hgf ()x = 4x 2 + px + q Find the value of p and the value of q. p = … q = … [4] Question 10 is printed on the next page.

13 marks

Mark scheme: 9(a) 3 1 9(b) 2 2 M1 for 2(1 − 2 x ) = x + 4 − oe 5 9(c) −2 x − 7 final answer 2 M1 for 1 – 2(x + 4) 9(d) 26 2 B1 for h(5) soi 2 or M1 for x 2 + 1 + 1 ( ) 9(e) 1 −x 2 M1 for x = 1 – 2y or 2x = 1 – y or oe final answer 2 y 1 = − x or y – 1 = – 2x 2 2 9(f) [p = ] – 20 4 B3 for [hgf(x)] = 4 x 2 − 20 x + 26 seen and not [q = ] 26 spoilt by further working or M1 for (1 – 2x) + 4 M1 dep for ( their (5 − 2 x ) ) 2 + 1 B1FT dep for 25 – 10x – 10x + 4x2

This question in 0580/42 Oct/Nov 2017

8 f ()x = 8 - 3x g (x) = , x ! - 1 h ()x = 2 x x + 1 (a) Find 8 (i) hf c 3 m, … [2] (ii) gh(-2), … [2] (iii) g -1 ()x , g -1 ()x = … [3] (iv) f -1 f (5 ) . … [1] (b) Write f(x) + g(x) as a single fraction in its simplest form. … [3]

11 marks

Mark scheme: 8(a)(i) 1 2 M1 for h(0) or for 28–3x 8(a)(ii) 8 2 10 M1 for g(¼) or for 2 x + 1 8(a)(iii) 10 −x 10 3 10 −y or − 1 final answer M2 for x = or better or x x y xy = 10 – x or better 10 or y + 1 = x or M1 for x(y + 1) = 10 or y(x + 1) = 10 10 10 or x = or x + 1 = y + 1 y 8(a)(iv) 5 1 8(b) −3 x 2 + 5 x + 18 3 (8 − 3 x )( x + 1) + 10 final answer M1 for x + 1 x + 1 B1 for – 3x2 – 3x + 8x + 8 [+10]

This question in 0580/42 May/June 2018

10 (a) f (x) = 2x - 3 g (x) = x 2 + 1 (i) Find gg(2). … [2] (ii) Find g (x + 2) , giving your answer in its simplest form. … [2] (iii) Find x when f ()x = 7 . x = … [2] (iv) Find f -1 ()x . f -1 ()x = … [2] (b) h (x) = x x , x 2 0 (i) Calculate h(0.3). Give your answer correct to 2 decimal places. … [2] (ii) Find x when h(x) = 256. x = … [1]

11 marks

Mark scheme: 10(a)(i) 26 2 2 2 M1 for g(5) or for x + 1 + 1 ( ) 10(a)(ii) x 2 + 4 x + 5 2 M1 for ( x + 2 ) 2 + 1 10(a)(iii) 5 2 M1 for 2 x − 3 = 7 10(a)(iv) x + 3 2 y 3 oe M1 for x = 2 y − 3 or y + 3 = 2 x or = x − oe 2 2 2 10(b)(i) [0].70 cao 2 B1 for [0].696 to [0].697 10(b)(ii) 4 cao 1

This question in 0580/43 May/June 2018

9 f (x) = 3x + 4 g (x) = 2x - 1 h ()x = 3 x 1 (a) Find ge 2 o. … [1] (b) Find fh (- 1) . … [2] (c) Find g -1 ()x . g -1 ()x = … [2] (d) Find ff ()x in its simplest form. … [2] (e) Find f ()x 2 in the form ax 2 + bx + c . ^ h … [2] (f) Find x when h -1 ( x) = g (2) . x = … [2]

11 marks

Mark scheme: 9(a) 0 1 9(b) 5 2 x 1 M1 for (33 ) + 4 or better or f ( ) or f (3–1) 3 9(c) x + 1 2 y 1 oe final answer M1 for x = 2 y − 1 or y + 1= 2 x or = x − 2 2 2 or better 9(d) 9 x + 16 2 M1 for 3(3x + 4) + 4 oe 9(e) 9 x 2 + 24 x + 16 2 B1 for three terms from 9 x 2 + 12 x + 12 x + 16 correct 9(f) 27 2 M1 for x = h(their g(2))

This question in 0580/43 Oct/Nov 2018

8 f (x) = , x ! - 2 g (x) = 8x - 5 h (x) = x 2 + 6 x + 2 1 (a) Work out g e 4 o. … [1] (b) Work out ff(2). … [2] (c) Find gg(x), giving your answer in its simplest form. … [2] (d) Find g -1 (x) . g -1 ( x) = … [2] (e) Write g (x) - f (x) as a single fraction in its simplest form. … [3] (f) (i) Show that hg(x) = 19 simplifies to 16x 2 - 20x + 3 = 0 . [3] (ii) Use the quadratic formula to solve 16x 2 - 20x + 3 = 0 . Show all your working and give your answers correct to 2 decimal places. x = … or x = … [4]

17 marks

Mark scheme: 8(a) −3 1 8(b) 12 2 3 oe M1 for soi 11 3 + 2 x + 2 8(c) 64 x − 45 final answer 2 M1 for 8 ( 8 x − 5 ) − 5 isw 8(d) x + 5 2 M1 for a correct first step y + 5 = 8 x , oe final answer 8 y 5 = x − or x = 8 y − 5 8 8 8(e) 8 x 2 + 11x − 13 3 M1 for ( 8 x − 5 )( x + 2 ) − 3 oe isw final answer x + 2 B1 for common denominator ( x + 2 ) 8(f)(i) ( 8 x − 5 ) 2 + 6 = 19 M1 64 x 2 − 40 x − 40 x + 25 B1 64 x 2 − 40 x − 40 x + 25 + 6 = 19 oe A1 with no errors and must show 2 leading to 16 x 2 − 20 x + 3 = 0 ( 8 x − 5 ) + 6 = 19 with no omissions after this 8(f)(ii) 2 2 2 [ −− ]20 ± ( [ − ]20 ) − 4 (16 )( 3 ) B1 for ( [ − ]20 ) − 4 (16 )( 3 ) or better oe 2 × 16 [ −− ]20 + q or B1 for oe or 2(16) [ −− ]20 − q 2(16) 0.17 and 1.08 final ans 2 B1 for each If 0 scored, SC1 for answer 0.2 and 1.1 or answer − 0.17 and −1.08 or 0.174... and 1.075 to 1.076 seen or 0.17 and 1.08 seen in working

This question in 0580/42 Feb/March 2019

9 f (x) = 7x - 2 g ( x) = x 2 + 1 h (x) = 3x (a) Find gh(2). … [2] (b) Find f – 1(x). f – 1(x) = … [2] (c) gg (x) = ax 4 + bx 2 + c Find the values of a, b and c. a = … b = … c = … [3] (d) Find x when hf(x) = 81. x = … [3]

10 marks

Mark scheme: 9(a) 82 2 M1 for (3x)2+1 soi by (32)2+1 or g(9) isw 9(b) x + 2 2 y 2 final answer M1 for y + 2 = 7x or = x − or 7 7 7 x = 7y – 2 9(c) [a =] 1, [b =] 2, [c =] 2 3 B2 for x 4 + x 2 + x 2 + 1 + 1 or M1 for ( x 2 + 1) 2 + 1 9(d) 6 3 M2 for 7x – 2 = 4 oe or M1 for 3x = 81 soi f(x) = 4 7 or for 37 x −=2 81 or better

This question in 0580/41 May/June 2019

(a) Use the graph to find (i) f (1 ) , … [1] (ii) ff (- 2) . … [2] (b) On the grid opposite, draw a suitable straight line to solve the equation 2 2 x - - 7 =- 3x for - 3 G x G 3 . x x = … or x = … [4] (c) By drawing a suitable tangent, find an estimate of the gradient of the curve at x = - 2. … [3] (d) (i) Complete the table for y = g (x) where g ()x = 2 -x for - 3 G x G 3 . x -3 -2 -1 0 1 2 3 y 2 1 0.5 0.125 [3] (ii) On the grid opposite, draw the graph of y = g (x) . [3] (iii) Use your graph to find the positive solution to the equation f (x) = g ( x) . x = … [1]

17 marks

Mark scheme: 5(a)(i) –3 1 5(a)(ii) 6.2 to 6.4 oe 2 M1 for 3 seen or used 5(b) y = 5 – 3x ruled 2 B1 for y = 5 – 3x soi or ruled line with gradient – 3 or with y – intercept at 5 (but not y = 5) or B1FT for incorrect line equation/expression shown in working and their line correctly drawn – 0.3 to – 0.2 2 B1 for each, dep on y = 5 – 3x drawn 1.65 to 1.8 or FT their line provided equation/expression shown in working, dep on B1FT for line 5(c) Tangent ruled at x = −2 1 B1 for correct tangent –4.5 to –2.5 2 Dep on B1 for tangent or close attempt at tangent at x = –2 M1 for rise/run also dep on tangent drawn or close attempt at correct tangent Must see correct or implied calculation from a drawn tangent 5(d)(i) 8, 4, 0.25 oe 3 B1 for each 5(d)(ii) Correct graph 3 B2FT for 6 or 7 correct plots or B1FT for 4 or 5 correct plots 5(d)(iii) 1.8 to 1.9 1

This question in 0580/43 May/June 2019

7 f (x) = 7 - 2x g ( x) = , x ! 0 h (x) = 27x x (a) Find (i) f (- 3 ) , … [1] (ii) hg ( 30) , … [2] (iii) f - 1 (x) . f - 1 (x) = … [2] (b) Solve. g (2x + 1) = 4 x = … [3] (c) Simplify, giving your answer as a single fraction. 1 + g (x) f (x) … [3] (d) Find h-1(19 683). … [1]

12 marks

Mark scheme: 7(a)(i) 13 1 7(a)(ii) 3 2 10  10  x M1 for h   oe soi or 27  30  7(a)(iii) 7 − x 2 M1 for x = 7 – 2y or y – 7 = –2x or 7 – y = 2x oe final answer y 7 2 or − = − + x oe 2 2 7(b) 0.75 oe final answer 3 10 M1 for = 4 2 x + 1 M1 for 10 = 8x + 4 or better 7(c) 70 − 19 x 70 − 19 x 3 M1 for x + 10(7 – 2x) or better isw or final 2 B1 for common denominator x(7 – 2x) oe isw x ( 7 − 2 x ) 7 x − 2 x answer 7(d) 3 final answer 1

This question in 0580/42 Oct/Nov 2019

9 f (x) = 2x - 3 g (x) = 9 - x 2 h (x) = 3x (a) Find (i) f (4 ) , … [1] (ii) hg ( 3) , … [2] (iii) g (2x) in its simplest form, … [1] (iv) fg (x) in its simplest form. … [2] (b) Find f - 1 ( x) . f - 1 ( x) = … [2] (c) Find x when 5f ()x = 3 . x = … [2] (d) Solve the equation gf (x) =- 16 . x = … or x = … [4] (e) Find x when h - 1 (x) =- 2 . x = … [1]

15 marks

Mark scheme: 9(a)(i) 5 1 9(a)(ii) 1 2 M1 for h(0) or 93 −x 2 or better 9(a)(iii) 9 – 4x2 final answer 1 9(a)(iv) 15 – 2x2 final answer 2 M1 for 2(9 – x2) –3 or better 9(b) x + 3 2 M1 for x = 2y – 3 or y + 3 = 2x or better final answer 2 y 3 or = x − 2 2 9(c) 4 9 2 3 1.8 or 15 or 5 M1 for 10 x − 15 = 3 or 2 x − 3 = 5 9(d) –1 and 4 nfww 4 M1 for 9 – (2x – 3)2= –16 A1 for 4x2 – 12x – 16[= 0] oe M1 (dep on first M1) for correct factors or use of formula or completing the square for their 3-term quadratic OR M1 for 9 – y2= –16 A1 for y2 = 25 M1 (dep on first M1) for 2x – 3= ±5 9(e) 1 1 9

This question in 0580/43 Oct/Nov 2019

10 f ( x) = 4 x - 1 g ( x) = x 2 h ( x) = 3 -x (a) Find in its simplest form (i) f ( x - 3) , … [1] (ii) g ( 5x) . … [1] (b) Find f -1 ( x) . f -1 ( x) = … [2] (c) Find the value of hh(l) , correct to 4 significant figures. … [3] (d) (i) Show that g ( 3x - 2) - h ( - 3) can be written as 9x 2 - 12x - 23 . [2] (ii) Use the quadratic formula to solve 9x 2 - 12 x - 23 = 0 . Give your answers correct to 2 decimal places. x = … or x = … [4] (e) Find x when f ( 61) = h ( x) . x = … [2]

15 marks

Mark scheme: 10(a)(i) 4 x − 13 final answer 1 10(a)(ii) 25x 2 final answer 1 10(b) x + 1 x 1 2 M1 for correct first step x = 4 y − 1 or or + 4 4 4 y 1 y + 1 = 4 x or = x − 4 4 10(c) 0.6934 final answer 3 1 3 or 0.693 B2 for 0.69336… or 3−oe or M1 for 3−3− x oe 10(d)(i) ( 3 x − 2 ) 2 − 3−−( 3) M1 9 x 2 − 6 x − 6 x + 4 − 27 or A1 with no errors seen 9 x 2 − 12 x + 4 − 27 leading to 9 x 2 − 12 x − 23 10(d)(ii) 2 B2 2 −−( 12) ± ( − 12) − 4(9)( −23) B1 for ( −12) − 4(9)( −23) oe 2 × 9 −−( 12) + q −−( 12) − q or better or oe or oe or 2 × 9 2 × 9 both – 1.07, 2.40 final answers B2 B1 for each If B0, SC1 for answers – 1.1 or –1.06 or –1.065… to – 1.065 and 2.4 or 2.39 or 2.398 to 2.398… or – 1.07 and 2.40 seen in working or for –2.40 and 1.07 as final answer 10(e) − 5 final answer 2 M1 for 243 = 3−x

This question in 0580/42 Feb/March 2020

6 f ( x) = 3x + 2 g ( x) = x 2 + 1 h ( x) = 4x (a) Find h(4). … [1] (b) Find fg(1). … [2] (c) Find gf(x) in the form ax 2 + bx + c . … [3] (d) Find x when f ( x) = g ( 7) . x = … [2] (e) Find f -1 ( x) . f -1 ( x) = … [2] g (x) (f) Find + x . f (x) Give your answer as a single fraction, in terms of x, in its simplest form. … [3] (g) Find x when h -1 ( x) = 2 . x = … [1]

14 marks

Mark scheme: 6(a) 256 1 6(b) 8 2 M1 for 3(x2 + 1) + 2 or for 3(2) + 2 6(c) 9 x 2 + 12 x + 5 3 M1 for (3x + 2)2 + 1 B1 for [(3x + 2)2 =] 9 x 2 + 6 x + 6 x + 4 oe 6(d) 16 2 M1 for 3x + 2 = 72 + 1 or better 6(e) x− 2 2 M1 for x = 3y + 2 or for y – 2 = 3x or for oe final answer y 2 3 = x + 3 3 6(f) 4 x 2 + 2 x + 1 3 B1 for x2 + 1 + x (3x + 2) or better seen final answer M1 for common denominator 3x + 2 3 x + 2 6(g) 16 1

This question in 0580/42 May/June 2020

11 f ( x) = 7 x - 4 g ( x) = , x ! 3 h ( x) = x2 x - 3 (a) Find g(6). … [1] (b) Find fg(4). … [2] (c) Find fh(x). … [1] f ( x) (d) Find + g ( x) . 2 Give your answer as a single fraction, in terms of x, in its simplest form. … [3] (e) Find the value of x when f ( x + 2) =- 11. x = … [2] (f) Find the values of p that satisfy h(p) = p. … [2]

11 marks

Mark scheme: 11(a) 4 1 11(b) 52 2 2 x M1 for f( 8 ) seen or 7 × − 4 x − 3 11(c) 7x2 – 4 1 11(d) 7 x 2 − 21x + 12 7 x 2 − 21x + 12 3 M1 for ( 7 x − 4 )( x − 3 ) + 2 × 2 x or 2( x − 3) 2 x − 6 B1 for denominator 2 ( x − 3 ) or 2x – 6 final answer 11(e) −3 2 M1 for 7 x + 14 − 4 = −11 11(f) [p =] 0 and [p =] 1 2 B1 for each

This question in 0580/43 May/June 2020

10 f ( x) = x 2 + 1 g ( x) = 1 - 2 x h ( x) = , x ! 0 j ( x) = 5x x (a) Find the value of (i) f(3), … [1] (ii) gf(3). … [1] (b) Find g -1 ( x) . g -1 ( x) = … [2] (c) Find x when h ( x) = 2 . x = … [1] (d) Find g(x)g(x) - gg(x), giving your answer in the form ax 2 + bx + c . … [4] (e) Find hh(x), giving your answer in its simplest form. … [1] (f) Find j(5). … [1] (g) Find x when j -1 ( x) = 2 . x = … [1] (h) j ( x) = hg ( - 12) Find the value of x. x = … [2] Question 11 is printed on the next page.

14 marks

Mark scheme: 10(a)(i) 10 1 10(a)(ii) –19 1 FT 1 – 2 their (a)(i) 10(b) 1 −x 2 y 1 oe final answer M1 for x = 1 – 2y or y + 2x = 1 or = − x 2 2 2 or y – 1 = –2x or better 10(c) 1 1 oe 2 10(d) 4x2 – 8x + 2 final answer 4 M1 for (1 – 2x)(1 – 2x) – (1– 2(1 – 2x)) or better B1 for 1 – 2x – 2x + 4x2 B1 for – (1 – 2 + 4x) or better or [+] 1 – 4x or for correct answer seen then spoiled 10(e) x final answer 1 10(f) 3125 1 10(g) 25 1 10(h) –2 2 1 B1 for or 0.04 25

This question in 0580/42 Oct/Nov 2020

10 f ( x) = 4 - 3 x g ( x) = x2 + x h ( x) = 3x (a) Find fh(2). … [2] (b) Find f -1 ( x) . f -1 ( x) = … [2] (c) Simplify. (i) f ( 1 - 2 x) … [2] (ii) gf ( x) - 9g ( x) … [4] 1 (d) = 9kx h ( x) Find the value of k. k = … [2]

12 marks

Mark scheme: 10(a) –23 2 M1 for 4 – 3(3x) oe soi 10(b) 4 − x 2 M1 for x = 4 – 3y or y + 3x = 4 oe final answer or x + 3y = 4 3 y 4 x 4 or = + x oe or = + y oe −3 −3 −3 −3 10(c)(i) 1 + 6x final answer 2 M1 for 4 – 3(1 – 2x) 10(c)(ii) 20 – 36x 4 B3 for 20 – 36x seen in working then or 4(5 – 9x) final answer spoiled OR M1 for (4 – 3x)2 + 4 – 3x – 9(x2 + x) or better B1 for [(4 – 3x)2=] 16 – 12x – 12x + 9x2 or better B1 for answer 20 – kx or k – 36x oe or answer 20 – 36x + kx2 k ≠ 0 10(d) 1 2 − 1 x 2 oe − oe M1 for (32)kx or 9kx = 9 2 11A 24 B1 5n – 1 oe B2 B1 for 5n – k or jn – 1 oe j ≠ 0 11B 127 B1 n3 + 2 oe B2 B1 for n3 oe 11C 256 B1 4(n–1) oe B2 B1 for 4k oe

This question in 0580/43 Oct/Nov 2020

12 f ( )x = 3 - 2 x g ( )x = x 2 + 5 h ( )x = x 3 (a) Find f ( - 5) . … [1] (b) Find ff(x). Give your answer in its simplest form. … [2] (c) Solve g ( x) = f ( x) + 37 . x = … or x = … [4] (d) Find f -1 ( )x . f -1 ( )x = … [2] (e) Find hf ( x) + g ( x) . Give your answer in its simplest form. … [5]

14 marks

Mark scheme: 12(a) 13 1 12(b) 4x – 3 final answer 2 M1 for 3 − 2 ( 3 − 2x ) 12(c) − 7 5 4 M1 for x 2 + 2 x − 35 [ = 0] or x 2 + 2 x = 35 M2 for ( x + 7 )( x − 5 ) or x ( x − 5 ) + 7 ( x − 5 ) or x ( x + 7 ) − 5 ( x + 7 ) or M1 for ( x + a )( x + b ) where a, b are integers with ab = − 35 or a + b = 2 12(d) 3 − x 2 M1 for a correct first step: oe final answer x = 3 − 2 y or y − 3 = −2 x , 2x = 3 – y or 2 y 3 = − x 2 2 12(e) 32 − 54 x + 37 x 2 − 8 x 3 5 B4 for final answer 27 − 36 x − 18 x + 24 x 2 + 12 x 2 − 8 x 3 + x2 + 5 oe OR 3 B1 for ( 3 − 2x ) + x2 + 5 and B2 for expansion of the 3 brackets, allow one error or B1 for correct expansion of 2 of the brackets with at least 3 terms correct

This question in 0580/41 May/June 2021

10 f ( )x = 3 x - 2 g ( )x = 5 x - 7 h ( )x = x 2 + x j ( )x = 3 x (a) Find (i) f(2), … [1] (ii) g(2), … [1] (iii) gf(2). … [1] (b) Find f -1 ( )x . f -1 ( )x = … [2] (c) Find hf(x), giving your answer in the form ax 2 + bx + c . … [3] (d) Find the derivative of h(x). … [1] (e) (i) Find x when j -1 ( )x = 4 . x = … [1] (ii) Simplify j -1 j ( )x . … [1]

11 marks

Mark scheme: 10(a)(i) 4 1 10(a)(ii) 3 1 10(a)(iii) 13 1 FT 5 × their (a)(i) – 7 10(b) x + 2 2 y 2 final answer M1 for y + 2 = 3x or for = x − 3 3 3 or for x = 3y – 2 10(c) 9x2 – 9x + 2 final answer 3 2 M1 for ( 3 x − 2 ) + 3 x − 2 2 2 B1 for ( 3 x − 2 ) = 9 x − 6 x − 6 x + 4 10(d) 2x + 1 1 10(e)(i) 81 1 10(e)(ii) x 1 Not y = x

This question in 0580/43 May/June 2021

8 (a) f ( )x = 3 - 5 x (i) Find x when f ( )x =- 5 . x = … [2] (ii) Find f - 1 ( )x . f - 1 ( )x = … [2] (b) g ( )x = 18 - 3x - x 2 (i) Write g ( )x in the form b - ( a + x) 2 . … [3] (ii) Sketch the graph of y = g ( x) . On your sketch, show the coordinates of the turning point. y O x [3] (iii) Find the equation of the tangent to the graph of y = 18 - 3 x - x 2 at x = 4 . Give your answer in the form y = mx + c . y = … [6]

16 marks

Mark scheme: 8(a)(i) 1.6 oe 2 M1 for 3 – 5x = – 5 8(a)(ii) 3 − x 2 y 3 oe final answer M1 for x = 3 – 5y or = − x or better, 5 5 5 or y – 3 = – 5x oe 8(b)(i) 2 3 Method 1 20.25 − (1.5 + x ) 2 B1 for ( ±1.5 ± x ) seen B1 for [b =] 18 + their 1.52 OR Method 2 B1 for b − a 2 − 2 ax − x 2 or for b = 20.25 B1 for a = 1.5 8(b)(ii) Correct sketch with max in correct 3 2 FT their 20.25 − ( their1.5 + x ) provided in quadrant at ( − 1.5, 20.25 ) that form B1 for ∩shape or for ∪shape if in form 2 c + ( d + x ) in part (b)(i) B1 for TP at ( − 1.5, k ) or ( k , 20.25 ) FT 2 their 20.25 ± ( their 1.5 + x ) or for (–1.5, 20.25) seen 8(b)(iii) [y =] 34 – 11x 6 B2 for –3 – 2x or B1 for either kx –3, k ≠ 0 or –2x + n or for 18 – 3 – 2x M1dep for gradient = their (–3 – 2(4) ) B1 for y-value at x = 4, is –10 M1dep for their –10 = (their –11)4 + c oe

This question in 0580/41 Oct/Nov 2021

11 f ( x) = 2 x - 1 g ( x) = x 2 + 2x h ( x) = 4x j ( x) = 2x (a) Find the value of (i) h(3), … [1] (ii) fh(3). … [1] (b) Solve the equation gf ( x) = 0 . x = … or x = … [4] (c) p -1 ( x) = f ( x) Find p(x). … [2] 1(d) h ( x) j ( x) = 2 Find the value of x. x = … [3]

11 marks

Mark scheme: 11(a)(i) 64 1 11(a)(ii) 127 1 FT 2 × their (a)(i) – 1 4 11(b) 1 M1 for ( 2 x − 1) 2 + 2(2 x − 1) ± oe nfww 2 2 B1 for 4 x − 2 x − 2 x + 1 or ( 2 x − 1)( 2 x −+1 2 ) B1 for 4 x 2 − 1 [= 0] or ( 2 x − 1)( 2 x + 1) [= 0] OR M1 for x(x + 2) = 0 (solving g(x) = 0) A1 for x = 0 or –2 B1 for 2x – 1 = 0 or 2x – 1 = –2 11(c) x + 1 2 M1 for oe final answer y 1 2 y + 1 = 2 x or = x − or x = 2 y − 1 2 2 11(d) 1 3 1 − oe nfww B2 for 3 x = − oe 6 2 OR 1 x M1 for 2 2 x × 2 x oe or 4 2 × 4 x oe or 8x oe 1 1 1 − − − M1 for 2 2 or 4 4 or 8 6 soi

This question in 0580/43 Oct/Nov 2021

3 f ( )x = 1 + 4 x g ( )x = x 2 (a) Find (i) gf(3), … [2] (ii) fg(x), … [1] (iii) f - 1 f ( )x . … [1] (b) Find the value of x when f ( )x = 15 . x = … [2]

6 marks

Mark scheme: 3(a)(i) 169 2 2 M1 for g(13) or (1 + 4 x ) or better 3(a)(ii) 1 + 4 x 2 final answer 1 3(a)(iii) x 1 3(b) 7 2 M1 for 1 + 4 x = 15 3.5 or 2

This question in 0580/42 Feb/March 2022

4 f ( x) = 2 x - 1 g ( x) = 3 x - 2 h ( x) = , x ! 0 j ( x) = 5x x (a) Find (i) f ( 2) , … [1] (ii) gf ( 2) . … [1] (b) Find g -1 ( x) . g -1 ( x) = … [2] (c) Find x when h ( x) = j (- 2) . x = … [2] (d) Write f ( x) - h ( x) as a single fraction. … [2] (e) Find the value of jj ( 2 ) . … [1] (f) Find x when j -1 ( x) = 4 . x = … [2]

11 marks

Mark scheme: 4(a)(i) 3 1 4(a)(ii) 7 1 FT their (i) 3×their (i) 2  4(b) 2 3  x oe final answer 2 M1 for y + 2 = 3x or 2 3 3   y x or x = 3y – 2 4(c) 25 2 M1 for 2 1 5  x oe 4(d) 2 2 1   x x x final answer 2 M1 for 2x – 1 – 1 x 4(e) 2.98 × 1017 or 2.980... × 1017 1 4(f) 625 2 M1 for x = j(4)

This question in 0580/41 May/June 2022

7 f ( x) = 10 - x g ( x) = , x ! 0 h ( x) = 2x j ( x) = 5 - 2 x x 1 (a) (i) Find g b 2 l. … [1] 1 (ii) Find hg b 2 l. … [1] (b) Find x when f ( x) = 7 . x = … [1] (c) Find x when g ( x) = h ( 3) . x = … [2] (d) Find j -1 ( x) . j -1 ( x) = … [2] (e) Write f ( x) + g ( x) + 1 as a single fraction in its simplest form. … [3] 2 2(f) f ( x) - ff ( x) = ax + bx + c ` j Find the values of a, b and c. a = … b = … c = … [4] (g) Find x when h -1 ( x) = 10 . x = … [2]

16 marks

Mark scheme: 7(a)(i) 4 1 7(a)(ii) 16 1 FT 2their 4 7(b) 3 1 7(c) 1 2 2 3 oe M1 for = 2 or better 4 x 7(d) 5 −x 2 M1 for oe final answer x = 5 – 2y or y + 2x = 5 oe 2 y 5 or = − x oe 2 2 7(e) 11x − x 2 + 2 3 x (10 − x ) + 2 + x final answer B2 for oe single fraction x x or B1 for x(10 – x) + 2 + x oe 2 or M1 for 10 − x + + 1 x 7(f) [a =] 1 4 B3 for x 2 − 21x + 100 [b =] –21 OR [c =] 100 2 M1 for (10 − x ) − (10 − (10 − x ) ) oe or better 2 2 B2 for [(10 − x ) ] = 100 − 10 x − 10 x + x or B1 for three out of four terms of [(10 − x ) 2 ] = 100 − 10 x − 10 x + x 2 correct 7(g) 1024 2 M1 for [x =] h(10) oe or better

This question in 0580/41 Oct/Nov 2022

11 f ( x) = 2 x - 1 g ( x) = 3x + 2 h ( x) = , x ! 0 j ( x) = x 2 x (a) Find j ( - 1) . … [1] (b) Find x when f ( x) + g ( x) = 0 . x = … [2] (c) Find gg(x), giving your answer in its simplest form. … [2] (d) Find hf ( x) + gh ( x) , giving your answer as a single fraction in its simplest form. … [4] (e) When pp ( x) = x, p ( x) is a function such that p -1 ( x) = p ( x) . Draw a ring around the function that has this property. 1 2 f ( x) = 2 x - 1 g ( x) = 3x + 2 h ( x) = , x ! 0 j ( x) = x x [1]

10 marks

Mark scheme: 11(a) 1 1 11(b) 1 2 M1 for 2x – 1 + 3x + 2 = 0 oe isw − or –0.2 5 11(c) 9x + 8 final answer 2 M1 for 3(3x + 2) + 2 11(d) 4 x 2 + 5 x − 3 4 final answer x (2 x − 1) 1  1 M1 for and 3 + 2 oe   2 x − 1  x  B1 for x + 3(2 x − 1) + 2 x (2 x − 1) oe or better isw B1 for common denominator = x(2x – 1) isw 4 x 2 + 9 x + 3 If 0 scored, SC1 for answer x (2 x + 1) 11(e) h(x) indicated 1

This question in 0580/42 Feb/March 2023

10 f ( )x = x - 4 g ( )x = 2 x + 5 h ( )x = 3 x (a) Find (i) f ( - 3) … [1] (ii) g -1 ( )x g -1 ( )x = … [2] (iii) f ( x) # g ( x) # f ( x) . … [4] (b) Find x when h ( x) = g ( f ( 2) ) . x = … [2]

9 marks

Mark scheme: 10(a)(i) −7 1 10(a)(ii) x  5 2 M1 for correct first step e.g. x  2 y  5 or oe final answer 2 2 x  y  5 y 5 or  x  2 2 10(a)(iii) 2 x 3  11x 2  8 x  80 final answer 4 M1 for  x  4   2 x  5   x  4  oe B2 for 2 x 3  8 x 2  8 x 2  5 x 2  20 x  20 x  32 x  80 or for simplified 4 term expression of the correct form with 3 terms correct in final answer or B1 for 3 terms correct out of 4 from x 2  4 x  4 x  16 or 2 x 2  8 x  5 x  20 10(b) 0 2 M1 for g(− 2) or 2(x – 4) + 5 oe or 3௫= 1 or g  f 2   1

This question in 0580/41 May/June 2023

9 f ( x) = ( 3 x + 1)( x + 5)( x - 4) g ( x) = 2x - 3 h ( x) = 4 2 x - 1 (a) Find (i) f ( 0) … [1] (ii) g -1 ( )x g -1 ( )x = … [2] (iii) gh(2). … [2] (b) g( 2)x = 7 Find the value of x. x = … [2] (c) Simplify g ( x 2 ) + gg ( x) + 1. … [3] (d) Find h -1 ( 16) . … [2] (e) f ( x) = ( 3 x + 1)( x + 5)( x - 4) This can be written in the form f ( x) = ax 3 + bx 2 + cx + d . Find the value of each of a, b, c and d. a = … b = … c = … d = … [3]

15 marks

Mark scheme: 9(a)(i) –20 1 9(a)(ii) x + 3 2 M1 for x = 2y – 3 or better or y + 3 = 2x or better oe final answer y 3 2 or = x – or better 2 2 9(a)(iii) 125 2 M1 for g(64) or 2(42x – 1) – 3 9(b) 2.5 oe 2 M1 for 2(2x) – 3 = 7 or better 9(c) 2x2 +4x – 11 final answer 3 B2 for 2x2 and either +4x or – 11 in final 3 term answer or for correct answer seen then spoiled or M1 for 2x2 – 3 + 2(2x – 3) – 3 [+ 1] 9(d) 1.5 oe 2 M1 for 42x – 1 = 42 or better 9(e) a = 3 3 B2 for 3 correct values b = 4 or for correct unsimplified expanded expression or c = –59 for simplified four-term expression of correct d = –20 form with 3 terms correct or B1 for 2 correct values or for correct expansion of one pair of brackets with at least 3 out of 4 terms correct.

This question in 0580/41 Oct/Nov 2023

11 f ( )x = 1 - 3 x g ( x) = ( x - 1) 2 h ( x) = , x ! 0 x (a) Find g(3). … [1] (b) Find f ( x - 2 ) , giving your answer in its simplest form. … [2] (c) Find f -1 ( )x . f -1 ( )x = … [2] (d) gf ( x) - g ( x) f ( x) = 3x 3 + ax 2 + bx + c Find the value of each of a, b and c. a = … b = … c = … [5] (e) Find h ( x) - f ( x) , giving your answer as a single fraction in its simplest form. … [3] (f) h ( x n ) = 3x 7 Find the value of n. n = … [1]

14 marks

Mark scheme: 11(a) 4 1 11(b) 7 – 3x final answer 2 M1 for 1 – 3(x – 2) 11(c) 1 −x 2 oe final answer 3 M1 for x = 1 – 3y or y – 1 = –3x or 1 – y y 1 = 3x or = − x 3 3 11(d) a = 2, b = 5, c = –1 5 B4 for two correct values only after correct substitution seen i.e. (1 – 3x – 1)2 – (x – 1)2(1 – 3x) or for correct unsimplified expansion or a correct simplified expansion. OR M1 for (1 – 3x – 1)2 – (x – 1)2(1 – 3x) B2 for correct expansion of [–](x – 1)2(1 – 3x) [–]( x2 – x – x + 1 – 3x3 + 3x2 + 3x2 – 3x) or better or B1 for expansion of one pair of brackets  ( x − 1) 2 =  x 2 − x − x + 1 or   better or [ (x – 1)(1 – 3x) =] – 3x2 + x + 3x – 1 11(e) 3 − x + 3 x 2 3 B1 for 3 − x(1 − 3 x) or better final answer x B1 for common denominator x isw 11(f) –7 1

This question in 0580/42 Oct/Nov 2023

6 f ( )x = 5 x - 3 g ( )x = 64 x h ( )x = , x !- 1 x + 1 (a) Find the value of (i) f ( 2) … [1] (ii) gf ( 0.5) . … [2] (b) Find h -1 ( )x . h -1 ( )x = … [3] 1 (c) Find x when g ( )x = 5 . 2 x = … [2] 1 (d) Write as a single fraction in its simplest form - h ( x) . f ( x) … [4]

12 marks

Mark scheme: 6(a)(i) 7 1 6(a)(ii) 1 2 M1 for g(–0.5) oe 5(x) – 3 or for 64 or better 8 6(b) 2 −x 2 3 2 or − 1 final answer M1 for y(x + 1) = 2 or x = or better x x y + 1 2 −y M1 for or xy = 2 – x oe y 6c 5 2 M1 for [64x =] 26x or (26)x or 6x = –5 − –0.833 or better 6 6(d) 7 − 9 x 7 − 9 x 4 1 2 or or B1 for − 2 (5 x − 3)( x + 1) 5 x + 2 x − 3 5 x − 3 x + 1 9 x − 7 − final answer M1 for x + 1 – 2 (5x – 3) seen isw 2 5 x + 2 x − 3 M1 for (5x – 3)(x + 1) seen isw

This question in 0580/43 Oct/Nov 2023

11 f ( x) = , x ! 0 g ( )x = 3 x - 5 h ( )x = 2 x x (a) Find. (i) gf(2) … [2] (ii) g -1 ( )x g -1 ( )x = … [2] (b) Find in its simplest form g ( x - 2) . … [2] (c) Find the value of x when (i) fg ( x) = 0.1 x = … [2] (ii) h ( x) - g ( 7) = 0 . x = … [2]

10 marks

Mark scheme: 11(a)(i) −3.5 oe 2  1   1 M1 for g   seen or 3  − 5 or better  2   x  11(a)(ii) x + 5 2 M1 for correct first step y + 5 = 3 x , oe final answer 3 y 5 = x − or x = 3 y − 5 3 3 11(b) 3x− 11 final answer 2 M1 for 3 ( x − 2 ) − 5 11(c)(i) 5 2 1 M1 for  = 0.1 3 x − 5 11(c)(ii) 4 nfww 2 M1 for 2 x − ( 3 −7 5 ) [ = 0] or better

This question in 0580/42 Feb/March 2024

7 (a) Solve 3x - 8 = 6 - 4x . x = … [2] (b) Factorise fully 10a 2 + 5a . … [2] (c) Factorise fully ( 2x - 3) 2 - 9 . … [2] 1 1 x (d) f ( )x = , x ! g ( )x = 3 4x - 1 4 (i) Find f ( 4) . … [1] (ii) Find gg ( 2) . … [2] (iii) Find k when g ( k) = f ( 7) . … [2]

11 marks

Mark scheme: 7(a) 2 2 M1 for 3 x  4 x  6  8 or better 7(b) 5a  2 a  1 final answer 2 B1 for a 10 a  5  or 5(2a2 +a) or 5a  2 a  1 then spoilt 7(c) 4 x  x  3  final answer 2 M1 for  (2 x  3)  3  (2 x  3)  3  or better or for 4 x 2  6 x  6 x  9 [ 9] oe or better 7(d)(i) 1 1 oe 15 7(d)(ii) 19 683 2 3 x B1 for g(9), 39 or 3 seen 7(d)(iii) −3 2 k 1 k 3 M1 for 3  or 3  3 27 or answer g(–3)

This question in 0580/42 May/June 2024

8 (a) f ( x) = 7 - 3 x g ( x) = x 2 - 16 (i) Find the values of x when g ( x) = 20 . x = … or x = … [2] (ii) Find f -1 ( )x . f -1 ( )x = … [2] (iii) Find gf ( x) + 1, giving your answer in its simplest form. … [3] (iv) On the axes, sketch the graph of y = g ( x) . On your sketch, indicate the values where the graph crosses the axes. y x O [4] (v) Find the equation of the tangent to the graph of y = g ( x) when x = -3. Give your answer in the form y = mx + c . y = … [5] (b) h ( x) = 3x (i) On the axes, sketch the graph of y = h ( x) . y x O [2] (ii) Write down the equation of the asymptote to the graph of y = h ( x) . … [1]

19 marks

Mark scheme: 8(a)(i) 6 and –6 2 M1 for x2 = 20 + 16 or better Or B1 for 6 or -6 8(a)(ii) 7 − x 2 y 7 oe final answer M1 for x = 7 – 3y or = − x 3 3 3 or y – 7 = – 3x oe or better 8(a)(iii) 9x2 – 42x + 34 final answer 3 M1 for (7 – 3x)2 – 16 [+ 1] oe B1 for 49 – 21x – 21x + 9x2 +k 8(a)(iv) Correct sketch with roots marked at –4 and 4 4 and y – intercept and turning point at y = – 16 B1 for correct parabola shape B2 for roots at –4 and 4 on graph and no extras or B1 for (x – 4) (x + 4) [= 0] or for one correct root on graph or for -4 and 4 seen B1 for turning point at (0, –16) 8(a)(v) [y =] – 6x – 25 5 M1 for derivative = 2x M1 for x = -3 substituted into their derivative B1 for (– 3, –7) soi M1 substitution of (– 3, their –7) into y = their –6x + c oe dep on 2nd M1 8(b)(i) Correct sketch with y – intercept above x – 2 axis B1 for correct shape 8(b)(ii) y = 0 1

This question in 0580/41 Oct/Nov 2024

9 f ( )x = 2x - 5 g ( )x = x 2 - 2x (a) Find (i) f ( 7 ) … [1] (ii) gf ( 7 ) … [1] (iii) f -1 ( )x . f -1 ( )x = … [2] (b) Find gf ( x) - 3g ( )x . Give your answer in the form ax 2 + bx + c . … [4]

8 marks

Mark scheme: 9(a)(i) 9 1 9(a)(ii) 63 1 FT (their (a)(i))2 – 2  their (a)(i) 9(a)(iii) x + 5 2 y 5 oe final answer M1 for x = 2y – 5 or y + 5 = 2x or = x – 2 2 2 9(b) x2 – 18x + 35 final answer 4 M1 for (2x – 5)2 – 2(2x – 5) – 3(x2 – 2x) B1 for 4x2 – 10x – 10x + 25 B1 for – 4x + 10 – 3x2 + 6x

This question in 0580/42 Oct/Nov 2024

11 f ( )x = 2x + 5 g ( )x = 1 - 2x h ( x) = , x ! -1 j ( )x = 2 x x + 1 (a) Find g(-3). … [1] (b) Find f ( x) g ( x) + fg ( x) + 1. Give your answer in its simplest form. … [4] (c) Find g -1 ( )x . g -1 ( )x = … [2] (d) Find hh(1). … [2] 1 (e) Simplify - h ( x) . f ( x) Give your answer as a single fraction in its simplest form. … [3] 1 (f) Find x when j ( )x = . 32 x = … [1] (g) Find x when j -1 ( )x = 0 . x = … [1]

14 marks

Mark scheme: 11(a) 7 1 11(b) −4 x 2 − 12 x + 13 final answer 4 B1 for (2x + 5)(1 – 2x) B1 for 2x – 4x2 + 5 – 10x oe B1 for 2(1 – 2x) + 5 11(c) 1 −x 2 M1 for oe final answer y 1 2 x = 1 − 2 y or 2 x = 1 − y or = − x 2 2 11(d) 2 2 oe 3  1  1 M1 for h   or oe  2  1 + 1 x + 1 11(e) −−x 4 −−x 4 3 M1 for x + 1 − (2 x + 5) oe or or (2 x + 5)( x + 1) 2 x 2 + 7 x + 5 x + 4 − 2 M1 for common denominator 2 x + 7 x + 5 (2 x + 5)( x + 1) seen final answer 11(f) –5 1 11(g) 1 1

This question in 0580/43 Oct/Nov 2024

14 f ( x) = 5 - 4x (a) Find f `– 3j. … [1] (b) Find f `3 - 2 xj. Give your answer in its simplest form. … [2] (c) Find f – 1 `xj. f – 1 `xj = … [2]

5 marks

Mark scheme: 14(a) 17 1 14(b) 8x – 7 final answer nfww 2 M1 for 5 – 4(3 – 2x) oe or better 14(c) 5 −x 2 y 5 oe final answer M1 for x = 5 – 4y or y – 5 = –4x or = − x 4 4 4 or better

This question in 0580/42 Feb/March 2025

19 f ( x) = x + 1 g ( x) = 5 - 2 x h ( x) = 2x (a) Find f ( –3) . … [1] (b) The domain of g ( x) is {–3, 0, 2}. Find the range of g ( x) . { … } [2] 1 (c) Find x when h ( x) = . 32 x = … [1] (d) Find x when h – 1 ( )x = 3 . x = … [2]

6 marks

Mark scheme: 19(a) –2 1 19(b) 11, 5, 1 2 B1 for 2 correct listed on answer line 19(c) –5 1 19(d) 8 2 M1 for [x =] h(3) or [x =] 23

This question in 0580/42 May/June 2025

28 f ( )x = 7 x - 4 Find the value of x when (a) f ( )x = 1 x = … [1] (b) f - 1 ( )x = 1. x = … [2]

3 marks

Mark scheme: 28(a) 4 1 28(b) 1 2 M1 for [x =] f(1) or better 343

This question in 0580/41 Oct/Nov 2025

19 f ( )x = 5 x g ( )x = 3x - 2 h ( )x = x 2 + 1 (a) Find f ( 5 ) . … [1] (b) Find g ( 8)x . … [1] (c) Find g -1 ( )x . g -1 ( )x = … [2] (d) Find the positive solution of gh ( )x = 364 . x = … [3] (e) Find ff -1 (12.) … [1]

8 marks

Mark scheme: 19(a) 3125 1 19(b) 24x – 2 or 2(12x – 1) final answer 1 19(c) x + 2 x 2 2 M1 for correct first step or + final answer 3 3 3 y 2 e.g. x = 3y – 2 or y + 2 = 3x or = x − 3 3 oe 19(d) 11 3 M2 for 3x2 + 1 = 364 or better or M1 for 3(x2 + 1) – 2 Alternative method: M2 for x2 + 1 = (364 + 2) ÷ 3 or M1 for 3x – 2 = 364 19(e) 12 1

This question in 0580/43 Oct/Nov 2025