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102 SEKOLAH BUKIT SION - IGCSE MATH REVISION
MATHEMATICS (EXTENDED) 0580 IGCSE MAY/JUNE 2017
REVISION 3
FUNCTIONS AND GRAPHS
103 SEKOLAH BUKIT SION - IGCSE MATH REVISION
NOTES:
104 SEKOLAH BUKIT SION - IGCSE MATH REVISION
EXERCISE A 1.
The diagram shows the graph of y = !
"+ "
!,for 0 < x < 8.
(a) Use the graph to solve the equation !
"+ "
!= 3.
Answer: x = …………… or x = …………… [2]
(b) By drawing a suitable tangent, work out an estimate of the gradient of the graph where x = 1.
Answer: ………………………………………… [3]
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2. (a) Find the co-ordinates of the midpoint of the line joining A(–8, 3) and B(–2, –3).
Answer: ( …………… , …………) [3]
(b) The line y = 4x + c passes through (2, 6). Find the value of c.
Answer: ………………………………………… [1]
(c) The lines 5x = 4y + 10 and 2y = kx – 4 are parallel.
Find the value of k.
Answer: ………………………………………… [2]
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3. f(x) = (x + 2)3 – 5 g(x) = 2x + 10 h(x) = (
" , x ≠ 0.
Find
(a) gf(x)
Answer: ………………………………………… [2]
(b) f -1(x)
Answer: ………………………………………… [3]
(c) gh (− (+)
Answer: ………………………………………… [2]
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4. The table shows some values for the equation y = x3 – 2x for –2 ≤ x ≤ 2.
(a) Complete the table of values. [3]
(b) On the grid below, draw the graph of y = x3 – 2x for –2 ≤ x ≤ 2. [4]
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(c) (i) On the grid, draw the line y = 0.8 for –2 ≤ x ≤ 2. [1] (ii) Use your graph to solve for the equation x3 – 2x = 0.8.
Answer: x = …………… or x = …………… or x = …………… [3]
(d) By drawing a suitable tangent, work out an estimate for the gradient of the graph of y= x3 – 2x where x = –1.5. You must show your working.
Answer: ………………………………………… [3] 5. Find the equation of the line passing through the points (0, -1) and (3, 5).
Answer: ………………………………………… [3]
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6. f(x) = 3x + 5 g(x) = 7 – 2x h(x) = x2 – 8 Find
(a) (i) f(3)
Answer: ………………………………………… [1] (ii) g(x – 3) in terms of x in its simplest form
Answer: ………………………………………… [2]
(iii) h(5x) in terms of x in its simplest form
Answer: ………………………………………… [1]
(b) Find the inverse function g -1(x).
Answer: ………………………………………… [2]
(c) Find hf(x) in the form of ax2 + bx + c.
Answer: ………………………………………… [3]
(d) Solve the equation ff(x) = 83.
Answer: ………………………………………… [3]
(e) Solve the inequality 2f(x) < g(x).
Answer: ………………………………………… [3]
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7. f(x) = 2x
(a) Complete the table. [3]
(b) Draw the graph of y = f(x) for 0 ≤ x ≤ 4. [4]
(c) Use your graph in part (b) to solve the equation 2x = 5.
Answer: ………………………………………… [1]
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(d) Draw a suitable straight line and use it to solve the equation 2x = 3x.
Answer: x =…………… or x = ……………… [3]
(e) Draw a suitable tangent and use it to find the co-ordinates of the point on the graph of y = f(x) where the gradient of the graph is 3.
Answer: ( …………… , …………) [3]
8. f(x) = 1 – 2x g(x) = (
" , x ≠ 0 h(x) = x3 + 1
(a) Find the value of
(i) gf(2)
Answer: ………………………………………… [2] (ii) h(–2)
Answer: ………………………………………… [1]
(b) Find fg(x). Write your answer as a single fraction.
Answer: ………………………………………… [2]
(c) Find h -1(x), the inverse of h(x).
Answer: ………………………………………… [2]
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(d) Write down which of these sketches shows the graph of each of y = f(x), y = g(x) and y = h(x).
Answer: y = f(x) Graph …………… y = g(x) Graph …………… y = h(x) Graph …………… [3]
(e) k(x) = x5 – 3 Solve the equation k -1(x) = 2.
Answer: ………………………………………… [2]
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9. f(x) = 5x + 4 g(x) = (!"
, x ≠ 0 h(x) = -(!."
Find
(a) fg(5),
Answer: ………………………………………… [2]
(b) gg(x) in its simplest form,
Answer: ………………………………………… [2]
(c) f -1(x)
Answer: ………………………………………… [2]
(d) the value of x when h(x) = 8.
Answer: ………………………………………… [2] 10. f(x) = x + !
" – 3, x ≠ 0 g(x) = "
!− 5
Find
(a) fg(18)
Answer: ………………………………………… [2]
(b) g-1(x)
Answer: ………………………………………… [2]
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11. f(x) = 3 – x – x2 g(x) = 3x
(a) Complete the tables of values for f(x) and g(x). [2]
(b) On the grid, draw the graphs of y = f(x) and g(x) for – 1.5 ≤ x ≤ 1.5. [6]
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(c) For – 1.5 ≤ x ≤ 1.5, use your graphs to solve (i) f(x) = 0
Answer: ………………………………………… [1] (ii) g(x) = 4
Answer: ………………………………………… [1]
(iii) fg(x) = g(x). Answer: ………………………………………… [1]
(d) By drawing a suitable tangent, find an estimate of the gradient of the graph of
y = f(x) when x = 0.5.
Answer: ………………………………………… [3] 12. f(x) = x2 + x – 3 g(x) = 2x + 7 h(x) = 2x
(a) Solve the equation f(x) = 0.
Show all your working and give your answers correct to 2 decimal places.
Answer: ………………………………………… [4]
(b) fg(x) = px2 + qx + r Find the values of p, q and r.
Answer: p = ……………………………… [3]
q = ……………………………… [3]
r = ……………………………… [3]
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(c) Find g -1(x).
Answer: ………………………………………… [2]
(d) Find x when h(x) = 0.25.
Answer: ………………………………………… [1]
(e) Find hhh(3).
Give your answer in standard form, correct to 4 significant figures.
Answer: ………………………………………… [4] 13. Find the equation of the line passing through the points with co-ordinates (5,9) and (−3,13).
Answer: ………………………………………… [3]
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14. The table shows some values for the function y = 11x – 2x2 – 12 for 1 ≤ x ≤ 4.5.
(a) Complete the table of values. [3]
(b) On the grid below, draw the graph of y = 11x – 2x2 – 12 for 1 ≤ x ≤ 4.5. [4]
(c) By drawing a suitable line, use your graph to solve the equation 11x – 2x2 = 11.
Answer: x = ……...… or x =………… [2]
(d) The line y = mx + 2 is a tangent to the curve y = 11x – 2x2 – 12 at the point P. By drawing this tangent, (i) find the co-ordinates of the point P
Answer: ( …………… , …………) [3] (ii) work out the value of m.
Answer: ………………………………………… [2]
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15. (a) Complete the table of values for the function f(x) = ("− 𝑥!, 𝑥 ≠ 0. [3]
(b) Draw the graph (next page) of f(x) = (
"− 𝑥! for −3 ≤ x ≤ –0.2 and 0.2 ≤ x ≤ 3. [5]
(c) Use your graph to solve f(x) = –3.
Answer: x = ……...… or x =………… or x =………… [3] (d) By drawing a suitable line on your graph, solve the equation f(x) = 2x – 2.
Answer: x = ……...… or x =………… or x =………… [3] (e) By drawing a suitable tangent, work out an estimate of the gradient of the curve at the point where x = –2. You must show working.
Answer: ………………………………………… [3]
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120 SEKOLAH BUKIT SION - IGCSE MATH REVISION
16.
A(5, 10) and B(13, −2) are two points on the line AB. The perpendicular bisector of the line AB has gradient !
3 .
Find the equation for the perpendicular bisector of AB.
Answer: ………………………………………… [4]
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17. The equation of the line l in the diagram is y = 5 – x.
(a) The line cuts the y-axis at P. Write down the co-ordinates of P.
Answer: ………………………………………… [1]
(b) Write down the gradient of the line l.
Answer: ………………………………………… [1]
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18. (a) Complete the table of values for y = x3 – 3x + 1. [2]
(b) Draw the graph (on the next page) of y = x3 – 3x + 1 for −2.5 ≤ x ≤ 2.5. [4] (c) By drawing a suitable tangent, estimate the gradient of the curve at the point where x = 2.
Answer: ………………………………………… [3] (d) Use your graph to solve the equation x3 – 3x + 1 = 1.
Answer: x = ……...… or x =………… x = ………… [2] (e) Use your graph to complete the inequality in k for which the equation
x3 – 3x + 1 = k has 3 different solutions.
Answer: …………… < k < ……………… [3]
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124 SEKOLAH BUKIT SION - IGCSE MATH REVISION
19. (a) f(x) = 2x – 3 g(x) = ("4(
+ 2 h(x) = 3x (i) Work out f(4).
Answer: ………………………………………… [1] (ii) Work out this fh(−1)
Answer: ………………………………………… [2] (iii) Find f-1(x), the inverse of f(x).
Answer: ………………………………………… [2] (iv) Find ff(x) in its simplest form.
Answer: ………………………………………… [2]
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(v) Show that the equation f(x) = g(x) simplifies to 2x2 – 3x – 6 = 0. [3] (vi) Solve the equation 2x2 – 3x – 6 = 0. Give your answers correct to 2 decimal places. Show all your working.
Answer: ………………………………………… [4]
(b) Simplify "673"4!"643"7(8
.
Answer: ………………………………………… [4]
126 SEKOLAH BUKIT SION - IGCSE MATH REVISION
20. f(x) = ("6−2x, x ≠ 0.
(a) Complete the table of values for f(x). [3]
(b) On the grid, draw the graph of y = f(x) for –3 ≤ x ≤ –0.5 and 0.4 ≤ x ≤ 2. [5]
(c) Solve the equation f(x) = 2.
Answer: ………………………………………… [1]
(d) Solve the equation f(x) = 2x + 3.
Answer: ………………………………………… [3]
(e) (i) Draw the tangent to the graph of y = f(x) at the point where x = –1.5. [1] (ii) Use the tangent to estimate the gradient of the graph of y = f(x) where x = –1.5.
Answer: ………………………………………… [2]
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EXERCISE B 21. f(x) = (
", 𝑥 ≠ 0 g(x) = 1 – x h(x) = x2 + 1
(a) Find fg((
!).
Answer: ………………………………………… [2]
(b) Find g-1(x), the inverse of g(x).
Answer: ………………………………………… [1]
(c) Find hg(x), giving your answers in simplest form.
Answer: ………………………………………… [3]
(d) Find the value of x when g(x) = 7.
Answer: ………………………………………… [1]
(e) Solve the solution h(x) = 3x. Show your working and give your answers correct to 2 decimal places.
Answer: ………………………………………… [4]
(f) A function k(x) is its own inverse where k-1(x) = k(x), For which of the functions f(x), g(x) and h(x) is this true?
Answer: ………………………………………… [1]
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2. The table shows some values for the function y = ("6+ 𝑥,𝑥 ≠ 0.
(a) Complete the table of values. [3] (b) On the grid, draw the graph of y = = (
"6+ 𝑥 for –3 ≤ x ≤ –0.5 and 0.5 ≤ x ≤ 4. [5]
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(c) Use your graph to solve the equation ("6+ 𝑥 − 3 = 0.
Answer: x = ……...… or x =………… x = ………… [3]
(d) Use your graph to solve the equation ("6+ 𝑥 = 1 − 𝑥.
Answer: ………………………………………… [3]
(e) By drawing a suitable tangent, find an estimate of the gradient of the curve at the point where x = 2.
Answer: ………………………………………… [3]
(f) Using algebra, show that you can use the graph at y = 0 to find √−1; . [3]
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3. f(x) = 4 – 3x g(x) = 3-x
(a) Find f(2x) in terms of x.
Answer: ………………………………………… [1]
(b) Find ff(x) in its simplest form.
Answer: ………………………………………… [2]
(c) Work out gg(–1). Give your answer as a fraction.
Answer: ………………………………………… [3]
(d) Find f-1(x), the inverse of f(x).
Answer: ………………………………………… [2]
(e) Solve the equation gf(x) = 1.
Answer: ………………………………………… [3]
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4. The point A has co-ordinates (−4, 6) and the point B has co-ordinates (7, −2). Calculate the length of the line AB.
Answer: ………………………………………… [3] 5. Given the function f(x) = 5 – 3x,
(a) Find f(6). Answer: ………………………………………… [1]
(b) Find f(x +2).
Answer: ………………………………………… [1]
(c) Find ff(x), in its simplest form.
Answer: ………………………………………… [2]
(d) Find f-1(x), the inverse of f(x).
Answer: ………………………………………… [2]
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6.
The diagram shows the straight line, l, which passes through (0, 3) and (4, 11).
(a) Find the equation of the line l in the form y = mx + c.
Answer: ………………………………………… [3]
(b) Line p is perpendicular to line l. Write down the gradient of line p.
Answer: ………………………………………… [1]
133 SEKOLAH BUKIT SION - IGCSE MATH REVISION
7. f(x) = x2 + 4x – 6
(a) f(x) can be written in the form (x + m)2 = n. Find the value of m and n.
Answer: m = …………………………… [1]
n = …………………………… [2]
(b) Use your answer in part (a) to find the positive solution to x2 + 4x – 6 = 0.
Answer: ………………………………………… [2]
134 SEKOLAH BUKIT SION - IGCSE MATH REVISION
8. f(x) = 3x + 5 g(x) = x2
(a) Find g(3x). Answer: ………………………………………… [1]
(b) Find f-1(x), the inverse function.
Answer: ………………………………………… [2]
(c) Find ff(x). Give your answer in its simplest form.
Answer: ………………………………………… [2] 9. The table shows some values for 𝑦 = 𝑥! − (
!", x ≠0.
(a) Complete the table of values. [3]
(b) On the grid, draw the graph of 𝑦 = 𝑥! − (!"
for −2 ≤ 𝑥 ≤ −0.2 and 0.2 ≤ 𝑥 ≤ 2 [4]
135 SEKOLAH BUKIT SION - IGCSE MATH REVISION
(c) By drawing a suitable line, use your graph to solve the equation 𝑥! − (!"= 2.
Answer: ………………………………………… [3]
(d) The equation 𝑥! − (!"= 𝑘 has only one solution.
Write down the range of values of k for which this is possible.
Answer: ………………………………………… [2]
(e) By drawing a suitable tangle, find an estimate of the gradient of the curve at the point where x = −1.
Answer: ………………………………………… [3]
136 SEKOLAH BUKIT SION - IGCSE MATH REVISION
10. y = x2 – 2x + 𝟏𝟐
𝒙, x ≠ 0.
(a) Complete the table of values below. [2]
(b) On the grid on the next page, draw the graph of y = x2 – 2x + (!"
for −4 ≤ 𝑥 ≤ −0.5 and 0.5 ≤ 𝑥 ≤ 4. [3]
(c) By drawing a suitable tangent, find an estimate of the gradient of the graph at the point (1, 11).
Answer: ………………………………………… [3]
(d) The equation x2 – 2x + 𝟏𝟐𝒙
= k has exactly two distinct solutions. Use the graph to find (i) the value of k
Answer: ………………………………………… [1]
(ii) the solutions of x2 – 2x + 𝟏𝟐𝒙
= k
Answer: ………………………………………… [2]
(e) The equation x2 + ax2 + bx + c = 0 can be solved by drawing the line y = 3x + 1 on the grid. Find the value of a, of b and of c.
Answer: ………………………………………… [3]
137 SEKOLAH BUKIT SION - IGCSE MATH REVISION
11. f(x) = B"6+ "
!, x ≠ 0.
(a) Complete the table of values for f(x). [3]
(b) On the grid, draw the graph of y = f(x) for −5 ≤ 𝑥 ≤ −1.5 and 1.5 ≤ 𝑥 ≤ −5. [5]
138 SEKOLAH BUKIT SION - IGCSE MATH REVISION
(c) Solve f(x) = 0.
Answer: ………………………………………… [1]
(d) By drawing a suitable line on the grid, solve the equation f(x) = 1 – x.
Answer: ………………………………………… [3]
(e) By drawing a tangent at the point (-3, -0.6), estimate the gradient of the graph of y = f(x) when x = −3.
Answer: ………………………………………… [3] 12. f(x) = 2x – 1 g(x) = x2 + x h(x) = !
", where x ≠ 0.
(a) Find ff(3).
Answer: ………………………………………… [2]
(b) Find gf(x), giving your answer in simplest form.
Answer: ………………………………………… [3]
(c) Find f-1(x).
Answer: ………………………………………… [2]
(d) Find h(x) + h(x + 2), giving your answer as a single fraction.
Answer: ………………………………………… [4]