Core Mathematics 3 Functions

Transcription

1 Core Mathematics 3 Functions Core Maths 3 Functions Page 1

2 Functions C3 The specifications suggest that you should be able to do the following: Understand the definition of a function, the domain and range and the inverse. Manipulate composite functions. Understand the modulus function and the transformations of f(x) to f(x) + a, f(ax), f(x - a), -f(x), f(-x), f( x ) and f(x). Core Maths 3 Functions Page 2

3 Functions A function is a mapping from one data set to another. A mapping is said to be one to one, two to one or many to one. A mapping is said to be a function if it is one to one (it also needs to be single valued and continuous) A function maps the values in the domain to the values in the range. In simple terms the Domain is the possible x values and the Range is the resulting y values. If we consider the graph of f(x) = x 2. y f(x) = x 2 x When x = 3 and -3 the values of f(x) are both 9 and therefore this mapping is said to be two to one. The mapping can be made into a function by restricting the domain Core Maths 3 Functions Page 3

4 Composite Functions We are asked to differentiate composite functions in C3 and a composite function is simply a function that is formed from two or three functions. For example if we have two functions f(x) = 3x + 2 and g(x) = e x then fg(x) and gf(x) are said to be composite functions. To find fg(x) you replace all x s in f(x) with the function g(x) Eg fg(x) = 3e x + 2 To find gf(x) you replace all x s in g(x) with the function f(x) Eg gf(x) = e 3x+2 The following example deals with solving equations where the function is composite. Example 1 The functions f and g are defined by 2 f : x x 2x 10 x 2 g : x x 4 where is a constant x a) Find the range of f b) Given that gf = 12, find the value of a) The range is the possible values of f(x) or the y values. We have been given a quadratic and therefore we need to complete the square to find the minima. To complete the square, remember to halve the coefficient of x. 2 2 x 2x 10 (x 1) 9 The quadratic has minima at x = 1 and y = 9.Therefore since x exists for all values of x the range is given by: f(x) 9 Core Maths 3 Functions Page 4

5 b) Given that gf = 12, find gf means substitute x = 4 into the function f and then substitute that value into g. f = = 18 gf = g(18) = Therefore = 12 = Finding the inverse. Example 2 x 4 f : x e - 8 x Find the inverse of f(x) and state its domain. The general rule for finding an inverse is to make x the subject and then swap the x and y around. x 4 y e - 8 y + 8 = e x 4 Taking ln of both sides: ln(y 8) x 4 x 4 ln(y 8) Swap x and y -1 f (x)= 4 ln(x 8) Core Maths 3 Functions Page 5

6 In graphical terms the inverse of a function is the reflection of the function in the line y = x. Therefore the domain and the range swap around. The graph below shows f(x). There is an asymptote at y = -8. Therefore the range of the function is f(x) > -8. This implies that the domain of f -1 (x) must be x > -8. Note that the asymptotes swap over from the function to its inverse. The second graph shows the reflection of f(x) in the line y = x (the inverse). Core Maths 3 Functions Page 6

7 Transformations When given a sketch of y = f(x), you need to be able to sketch transformations of the graph, showing coordinates of the points to which the given points are mapped. These may be combined to give, for example bf(x + a), which is a horizontal translation of a followed by a vertical stretch of scale factor b. For combinations of transformations, the graph can be built up one step at a time starting from a given curve. Core Maths 3 Functions Page 7

8 The modulus function You will be asked to sketch the graphs of y = f( x ) and y = f(x) and solve equations of the type f( x ) = a and f(x) = b. The examples below cover most eventualities. Example 3 The functions f and g are defined by f : x x a - a x Where a is a constant. g : x 4x a x a) Use algebra to find, in terms of a, the coordinates of the point at which the graphs of f and g intersect. b) Find an expression for fg(x). c) Solve, for x in terms of a, the equation fg(x) = 4a a) To sketch f(x) we first need to consider the graph of y = x a, secondly y = x-a and finally y = x - a - a. The first is straight from year 8 work. The second takes the line and reflects the negative part above the x axis. The final transformation is to translate the line down a. Core Maths 3 Functions Page 8

9 y = x - a y = x - a a - a a - a y = x - a - a a - a The second function g(x) is simply a straight line of gradient 4 and intercept a. y = 4x + a y = x - a - a a - a Note that there is only one point of intersection between f(x) and g(x), but the reflected line is y = -x + a - a Therefore equating the two gives: -x = 4x + a x = - 0.2a and hence y = 0.2a Core Maths 3 Functions Page 9

10 b) fg(x) means take g(x) and swap all x s in f(x) with g(x). fg(x) 4x a a - a fg(x) = 4x a Solving for fg(x) = 4a gives: 4x a 4a 4x 5a 5 x a 4 Example 4 The functions f is defined by f : x 2x 4-4 x a) Solve the equation f(x) = 2. The equation g is defined by 2 g : x x - 4x + 12 x 2 b) Find the range of g. c) Find gf(-5) Core Maths 3 Functions Page 10

11 a) The example above detailed the build up of a modulus graph and the same idea is used to display the graph below. As can be seen from the graph there are two solutions to f(x) = 2. The line to the left has equation y = - 2x. The equation to the right has equation y = 2x 4 4 = 2x - 8 Therefore solving y = 2 for both parts gives: 2 = -2x and 2x 8 = 2-1 = x x = 5 b) Completing the square for g(x) gives: (x 2) Therefore g(x) has a minima at x = 2 and is always greater than 8. Hence the range is g(x) 8. Core Maths 3 Functions Page 11

12 c) gf(-5) f(-5) = = 10 Therefore gf(-5) = = 72 The examples given above have detailed y = f(x) where the graph of f(x) is reflected in the x axis. y = f( x ) is somewhat different. The example below should clarify the difference Example 5 y 4 B( 3, 1) 1 O A(2, 0) 3 x The diagram above shows a sketch of the curve with equation y = f(x), -1 x 3. The curve touches the x-axis at the origin O, crosses the 4 x-axis at the point A(2, 0) and has a maximum at the point B(, 1). 3 In separate diagrams, show a sketch of the curve with equation (a) y = f(x + 1), (b) y = f(x), (c) y = f( x ), marking on each sketch the coordinates of points at which the curve Core Maths 3 Functions Page 12

13 (i) has a turning point, (ii) meets the x-axis. a) y = f(x + 1): is simply a translation of one unit to the left. y 1 (, 1) x b) y = f(x) : reflect the graph in the x axis. y 4 (, 1) x c) y = f( x ): only consider positive values of x and reflect the curve in the y axis. 4 (, 1) 3 y 4 (, 1) x Core Maths 3 Functions Page 13

14 Edexcel C3 Function past paper questions section_01 1. The function f is defined by 5x 1 3 f: x, x > 1. 2 x x 2 x 2 2 (a) Show that f(x) =, x > 1. x 1 (b) Find f 1 (x). The function g is defined by (c) Solve fg(x) = 4 1. g: x x 2 + 5, x R. 2. The functions f and g are defined by f : x 2x + ln 2, x R, (Q3, June 2005) g : x e 2x, x R. (a) Prove that the composite function gf is gf : x 4e 4x, x R. (b) Sketch the curve with equation y = gf(x), and show the coordinates of the point where the curve cuts the y-axis. (c) Write down the range of gf. (d) Find the value of x for which d [gf(x)] = 3, giving your answer to 3 significant figures. dx (Q8, Jan 2006) 3. For the constant k, where k > 1, the functions f and g are defined by f: x ln (x + k), x > k, g: x 2x k, x R. (a) On separate axes, sketch the graph of f and the graph of g. On each sketch state, in terms of k, the coordinates of points where the graph meets the coordinate axes. (5) (b) Write down the range of f. k (c) Find fg in terms of k, giving your answer in its simplest form. 4 Core Maths 3 Functions Page 14

15 The curve C has equation y = f(x). The tangent to C at the point with x-coordinate 3 is parallel to the line with equation 9y = 2x + 1. (d) Find the value of k. (Q7, June 2006) 4. The function f is defined by f : x ln (4 2x), x < 2 and x R. (a) Show that the inverse function of f is defined by f 1 : x e x and write down the domain of f 1. (b) Write down the range of f 1. (c) Sketch the graph of y = f 1 (x). State the coordinates of the points of intersection with the x and y axes. (Q6, Jan 2007) 5. The functions f and g are defined by f : ln (2x 1), x R, x > 1 2, 2 g :, x R, x 3. x 3 (a) Find the exact value of fg. (b) Find the inverse function f 1 (x), stating its domain. (c) Sketch the graph of y = g(x). Indicate clearly the equation of the vertical asymptote and the coordinates of the point at which the graph crosses the y-axis. 2 (d) Find the exact values of x for which = 3. x 3 (Q5, June 2007) 6. The functions f and g are defined by f : x 1 2x 3, x R. g : x x 3 4, x > 0, x R. (a) Find the inverse function f 1. (b) Show that the composite function gf is 3 8x 1 gf : x x Core Maths 3 Functions Page 15

16 (c) Solve gf (x) = 0. (d) Use calculus to find the coordinates of the stationary point on the graph of y = gf(x). (5) (Q8, Jan 2008) 7. The function f is defined by 2( x 1) f: x 2 x 2x 3 1 (a) Show that f(x) =, x > 3. x 1 1, x > 3. x 3 (b) Find the range of f. (c) Find f 1 (x). State the domain of this inverse function. The function g is defined by g: x 2x 2 3, x R. (d) Solve fg(x) = The functions f and g are defined by f : x 3x + ln x, x > 0, x R, (a) Write down the range of g. g : x 2 e x, x R. (b) Show that the composite function fg is defined by (c) Write down the range of fg. d dx (d) Solve the equation fg( x) fg : x x 2 + = x( x e x2 + 2). 2 3e x, x R. (Q4, June 2008) (6) (Q5, Jan 2009) 9. (i) Find the exact solutions to the equations (a) ln (3x 7) = 5, (b) 3 x e 7x + 2 = 15. (ii) The functions f and g are defined by f (x) = e 2x + 3, x R, g(x) = ln (x 1), x R, x > 1. (5) Core Maths 3 Functions Page 16

17 (a) Find f 1 and state its domain. (b) Find fg and state its range. (Q9, Jan 2010) 10. The function f is defined by f : x 2x 5, x R. (a) Sketch the graph with equation y = f(x), showing the coordinates of the points where the graph cuts or meets the axes. (b) Solve f(x) =15 + x. The function g is defined by g : x x 2 4x + 1, x R, 0 x 5. (c) Find fg. (d) Find the range of g. (Q4, June 2010) 11. The function f is defined by (a) Find f 1 (x). f: x 3 2x, x R, x 5. x 5 Core Maths 3 Functions Page 17

18 The function g has domain 1 x 8, and is linear from ( 1, 9) to (2, 0) and from (2, 0) to (8, 4). Figure 2 shows a sketch of the graph of y = g(x) (b) Write down the range of g. (c) Find gg. (d) Find fg(8). (e) On separate diagrams, sketch the graph with equation (i) y = g(x), (ii) y = g 1 (x). Show on each sketch the coordinates of each point at which the graph meets or cuts the axes. (f) State the domain of the inverse function g 1. (Q6, Jan 2011) 12. The function f is defined by f : x 4 ln (x + 2), x R, x 1. (a) Find f 1 (x). (b) Find the domain of f 1. The function g is defined by (c) g : x e x 2 2, x R. Find fg(x), giving your answer in its simplest form. (d) Find the range of fg. 13. The function f is defined by (Q4, June 2011) f : x 3( x 1) 2 2x 7x 4 1 1, x R, x >. x 4 2 (a) Show that f(x) = (b) Find f 1 (x). 1. 2x 1 (c) Find the domain of f 1. g(x) = ln (x + 1). (d) Find the solution of fg(x) = 7 1, giving your answer in terms of e. (Q7, Jan 2012) Core Maths 3 Functions Page 18

19 14. The functions f and g are defined by f: x e x + 2, x R, g : x ln x, x > 0. (a) State the range of f. (b) Find fg(x), giving your answer in its simplest form. (c) Find the exact value of x for which f(2x + 3) = 6. (d) Find f 1, the inverse function of f, stating its domain. (e) On the same axes sketch the curves with equation y = f(x) and y = f 1 (x), giving the coordinates of all the points where the curves cross the axes. (Q6, June 2012) 15. h(x) = 2 x x 2 5 ( x 2 18, x 0. 5)( x 2) (a) Show that h(x) = 2x. x 2 5 (b) Hence, or otherwise, find h (x) in its simplest form. Figure 2 Figure 2 shows a graph of the curve with equation y = h(x). (c) Calculate the range of h(x). Core Maths 3 Functions Page 19

20 (5) (Q7, Jan 2013) 16. The function f has domain 2 x 6 and is linear from ( 2, 10) to (2, 0) and from (2, 0) to (6, 4). A sketch of the graph of y = f(x) is shown in Figure 1. Figure 1 (a) Write down the range of f. (b) Find ff(0). The function g is defined by g : 4 3x x, x R, x 5 5 x (c) Find g 1 (x) (d) Solve the equation gf(x) = 16 (5) (Q7, June 2013) Core Maths 3 Functions Page 20

21 17. The functions f and g are defined by f : x 2 x 3, x R g : x 3 4x, x R 18. (a) State the range of f. (b) Find fg. (c) Find g 1, the inverse function of g. (d) Solve the equation gg(x) + [g(x)] 2 = 0 x 3 2x 1 g( x) 2 x 3 x x 6, x > 3 x 1 (a) Show that g( x), x > 3 x 2 (b) Find the range of g. (c) Find the exact value of a for which g(a) = g 1 (a). (5) (Q7, June 2013_R) (Q5, June 2013) 19. The function f is defined by f : 2 2 e x k x, x R, k is a positive constant. (a) State the range of f. (b) Find f 1 and state its domain. The function g is defined by g : x ln 2x, x > 0 Core Maths 3 Functions Page 21

22 (c) Solve the equation g(x) + g(x 2 ) + g(x 3 ) = giving your answer in its simplest form. (d) Find fg(x), giving your answer in its simplest form. (e) Find, in terms of the constant k, the solution of the equation fg(x) = 2k 2 (Q6, June 2014_R) Figure 2 Figure 2 shows a sketch of part of the curve with equation g(x) = x 2 (1 x)e 2x, x 0. (a) Show that g' (x) = f(x)e 2x, where f(x) is a cubic function to be found. (b) Hence find the range of g. (c) State a reason why the function g 1 (x) does not exist. (6) (Q7, June 2015) Core Maths 3 Functions Page 22

23 DO NOT WRITE IN THIS A 21. The functions f and g are defined by f : x 7x 1, x R, g : x 4, x 2, x R, x (a) Solve the equation fg(x) = x. (b) Hence, or otherwise, find the largest value of a such that g(a) = f 1 (a). (Q1, June 2016) y blank O Figure 1 x Figure 1 shows a sketch of part of the graph of y = g(x), where g(x) = 3 + x + 2, x 2 (a) State the range of g. (b) Find g 1 (x) and state its domain. (c) Find the exact value of x for which g(x) = x (d) Hence state the value of a for which g(a) = g 1 (a) (Q3, June 2017) Core Maths 3 Functions Page 23

24 23. The function f is defined by 3x 5 f: x x 1, x, x 1 (a) Find an expression for f 1 (x). (b) Show that ff x where a is an integer to be determined. The function g is defined by x a, x, x 1, x 1 x 1 g : x 2 x 3x, x,0 x 5 (c) Find the value of fg. (d) Find the range of g. 24. The function g is defined by (Q7, IAL, June 2014) g : x 8 2x, x R, x 0 (a) Sketch the graph with equation y = g(x), showing the coordinates of the points where the graph cuts or meets the axes. (b) Solve the equation 8 2x = x + 5 The function f is defined by f : x x 2 3x + 1, x R, 0 x 4 (c) Find fg(5). (d) Find the range of f. You must make your method clear. (Q7, IAL, Jan 2015) Core Maths 3 Functions Page 24

25 25. Figure 4 Figure 4 shows a sketch of part of the curve with equation y = f(x), x The curve meets the coordinate axes at the points A(0, 3) and B( 1 ln4, 0) and the 3 curve has an asymptote with equation y = 4 In separate diagrams, sketch the graph with equation (a) y = f x (b) y = 2f(x) + 6 On each sketch, give the exact coordinates of the points where the curve crosses or meets the coordinate axes and the equation of any asymptote. Given that f(x) = e 3x 4, x 1 g(x) = ln x 2, x > 2 (c) state the range of f, (d) find f 1 (x), (e) express fg(x) as a polynomial in x. (Q11, IAL, Jan 2016) Core Maths 3 Functions Page 25

26 26. Given that f(x) = 4 3x + 5, x > 0 g(x) = 1 x, x > 0 (a) state the range of f, (b) find f 1 (x), (c) find fg(x). (d) Show that the equation fg(x) = gf(x) has no real solutions. 27. The function g is defined by (a) Find the range of g. g (x) = (b) Find g 1 (x) and state its domain. 6x 2x + 3 x > 0 (Q4, IAL, Jan 2017) (c) Find the function gg(x), writing your answer as a single fraction in its simplest form. (Q3, IAL, June 2017) Core Maths 3 Functions Page 26

27 Edexcel C3 Function past paper questions section_02 1. Figure 1 y 1 O 3 b x (1, a) Figure 1 shows part of the graph of y = f(x), x R. The graph consists of two line segments that meet at the point (1, a), a < 0. One line meets the x-axis at (3, 0). The other line meets the x-axis at ( 1, 0) and the y-axis at (0, b), b < 0. In separate diagrams, sketch the graph with equation (a) y = f(x + 1), (b) y = f( x ). Indicate clearly on each sketch the coordinates of any points of intersection with the axes. Given that f(x) = x 1 2, find (c) the value of a and the value of b, (d) the value of x for which f(x) = 5x. (Q6, June 2005) Core Maths 3 Functions Page 27

28 2. Figure 1 y M (2, 4) 5 O 5 x Figure 1 shows the graph of y = f(x), 5 x 5. The point M (2, 4) is the maximum turning point of the graph. Sketch, on separate diagrams, the graphs of (a) y = f(x) + 3, (b) y = f(x), (c) y = f( x ). Show on each graph the coordinates of any maximum turning points. (Q1, Jan 2006) Core Maths 3 Functions Page 28

29 3. Figure 1 y y = f(x) O Q (3, 0) x (0, 2) P Figure 1 shows part of the curve with equation y = f(x), x R, where f is an increasing function of x. The curve passes through the points P(0, 2) and Q(3, 0) as shown. In separate diagrams, sketch the curve with equation (a) y = f(x), (b) y = f 1 (x), (c) y = 21 f(3x). Indicate clearly on each sketch the coordinates of the points at which the curve crosses or meets the axes. (Q3, June 2006) Core Maths 3 Functions Page 29

30 4. Figure 1 Figure 1 shows a sketch of the curve with equation y = f (x). The curve passes through the origin O and the points A(5, 4) and B( 5, 4). In separate diagrams, sketch the graph with equation (a) y = f (x), (b) y = f ( x ), (c) y = 2f(x + 1). On each sketch, show the coordinates of the points corresponding to A and B. (Q4, Jan 2008) Core Maths 3 Functions Page 30

31 5. Figure 1 Figure 1 shows the graph of y = f(x), x R, The graph consists of two line segments that meet at the point P. The graph cuts the y-axis at the point Q and the x-axis at the points ( 3, 0) and R. Sketch, on separate diagrams, the graphs of (a) y = f(x), (b) y = f ( x). Given that f(x) = 2 x + 1, (c) find the coordinates of the points P, Q and R, (d) solve f(x) = 21 x. (5) (Q3, Jan 2008) Core Maths 3 Functions Page 31

32 6. Figure 1 Figure 1 shows the graph of y = f (x), 1 < x < 9. The points T(3, 5) and S(7, 2) are turning points on the graph. Sketch, on separate diagrams, the graphs of (a) y = 2f(x) 4, (b) y = f(x). Indicate on each diagram the coordinates of any turning points on your sketch. (Q3, Jan 2009) Core Maths 3 Functions Page 32

33 7. Figure 2 shows a sketch of part of the curve with equation y = f(x), x R. The curve meets the coordinate axes at the points A(0, 1 k) and B( 21 ln k, 0), where k is a constant and k >1, as shown in Figure 2. On separate diagrams, sketch the curve with equation (a) y = f(x), (b) y = f 1 (x). Show on each sketch the coordinates, in terms of k, of each point at which the curve meets or cuts the axes. Given that f(x) = e 2x k, (c) state the range of f, (d) find f 1 (x), (e) write down the domain of f 1. (Q5, June 2009) Core Maths 3 Functions Page 33

34 8. Figure 1 Figure 1 shows a sketch of the graph of y = f (x). The graph intersects the y-axis at the point (0, 1) and the point A(2, 3) is the maximum turning point. Sketch, on separate axes, the graphs of (i) y = f( x) + 1, (ii) y = f(x + 2) + 3, (iii) y = 2f(2x). On each sketch, show the coordinates of the point at which your graph intersects the y-axis and the coordinates of the point to which A is transformed. (9) (Q6, Jan 2010) Core Maths 3 Functions Page 34

35 9. The curve has a turning point at A(3, 4) and also passes through the point (0, 5). (a) Write down the coordinates of the point to which A is transformed on the curve with equation (i) y = f(x), (ii) y = 2f( 21 x). (b) Sketch the curve with equation y = f( x ). On your sketch show the coordinates of all turning points and the coordinates of the point at which the curve cuts the y-axis. The curve with equation y = f(x) is a translation of the curve with equation y = x 2. (c) Find f(x). (d) Explain why the function f does not have an inverse. (Q6, June 2010) Core Maths 3 Functions Page 35

36 10. Figure 1 Figure 1 shows part of the graph of y = f (x), x R. The graph consists of two line segments that meet at the point R (4, 3), as shown in Figure 1. Sketch, on separate diagrams, the graphs of (a) y = 2f(x + 4), (b) y = f( x). On each diagram, show the coordinates of the point corresponding to R. (Q3, June 2011) Core Maths 3 Functions Page 36

37 11. Figure 1 Figure 1 shows the graph of equation y = f(x). The points P ( 3, 0) and Q (2, 4) are stationary points on the graph. Sketch, on separate diagrams, the graphs of (a) y = 3f(x + 2), (b) y = f(x). On each diagram, show the coordinates of any stationary points. (Q2, Jan 2012) Core Maths 3 Functions Page 37

38 12. Figure 2 Figure 2 shows part of the curve with equation y = f(x). The curve passes through the points P( 1.5, 0)and Q(0, 5) as shown. On separate diagrams, sketch the curve with equation (a) y = f(x) (b) y = f( x ) (c) y = 2f(3x) Indicate clearly on each sketch the coordinates of the points at which the curve crosses or meets the axes. (Q4, June 2012) Core Maths 3 Functions Page 38

39 13. Figure 1 Figure 1 shows part of the curve with equation y = f(x), x R. The curve passes through the points Q(0, 2) and P( 3, 0) as shown. (a) Find the value of ff ( 3). On separate diagrams, sketch the curve with equation (b) y = f 1 (x), (c) y = f( x ) 2, (d) y = 2f 1 2 x. Indicate clearly on each sketch the coordinates of the points at which the curve crosses or meets the axes. (Q3, Jan 2013) 14. Given that f(x) = ln x, x > 0 sketch on separate axes the graphs of (i) y = f(x), (ii) y = f(x), (iii) y = f(x 4). Show, on each diagram, the point where the graph meets or crosses the x-axis. In each case, state the equation of the asymptote. (7) (Q2, June 2013) Core Maths 3 Functions Page 39

40 15. Figure 1 Figure 1 shows a sketch of the curve with equation y = f(x), x > 0, where f is an increasing function of x. The curve crosses the x-axis at the point (1, 0) and the line x = 0 is an asymptote to the curve. On separate diagrams, sketch the curve with equation (a) y = f(2x), x > 0 (b) y = f(x), x > Indicate clearly on each sketch the coordinates of the point at which the curve crosses or meets the x-axis. (Q2, June 2013_R) Figure 1 Figure 1 shows part of the graph with equation y = f (x), x R. The graph consists of two line segments that meet at the point Q(6, 1). Core Maths 3 Functions Page 40

41 The graph crosses the y-axis at the point P(0, 11). Sketch, on separate diagrams, the graphs of (a) y = f (x) (b) y = 2f ( x) + 3 On each diagram, show the coordinates of the points corresponding to P and Q. Given that f (x) = a x b 1, where a and b are constants, (c) state the value of a and the value of b. (Q4, June 2014) 17. (a) Sketch the graph with equation y = 4x 3 stating the coordinates of any points where the graph cuts or meets the axes. Find the complete set of values of x for which (b) (c) 18. Given that 4x 3 > 2 2x 4x 3 > 3 2x 2 (Q5, June 2014_R) f(x) = 2e x 5, x R, (a) sketch, on separate diagrams, the curve with equation (i) y = f (x), (ii) y = f (x). On each diagram, show the coordinates of each point at which the curve meets or cuts the axes. On each diagram state the equation of the asymptote. (b) Deduce the set of values of x for which f (x) = f (x). (c) Find the exact solutions of the equation f (x) = 2. (6) (Q2, June 2015) Core Maths 3 Functions Page 41

42 19. Given that a and b are positive constants, (a) on separate diagrams, sketch the graph with equation (i) y = 2x a (ii) y = 2x a + b Show, on each sketch, the coordinates of each point at which the graph crosses or meets the axes. Given that the equation 2x a + b = 3 2 x + 8 has a solution at x = 0 and a solution at x = c, (b) find c in terms of a. 20. Given that a and b are constants and that a > b > 0 (Q6, June 2017) (a) on separate diagrams, sketch the graph with equation (i) y = x a (ii) y = x a b Show on each sketch the coordinates of each point at which the graph crosses or meets the x-axis and the y-axis. (5) (b) Hence or otherwise find the complete set of values of x for which x a b < 1 2 x giving your answer in terms of a and b. (Q6, IAL. June 2016) Core Maths 3 Functions Page 42

43 DO NOT WRITE IN THIS AR 21. y 5 Q O P 1 3 Figure 2 x Figure 2 shows a sketch of the graph of y = f (x), x R. 1 The point P,0 is the vertex of the graph. 3 The point Q (0, 5) is the intercept with the y-axis. Given that f (x) = ax b, where a and b are constants, (a) (i) find all possible values for a and b, (ii) hence find an equation for the graph. (b) Sketch the graph with equation 1 y f x 3 2 showing the coordinates of its vertex and its intercept with the y-axis. (Q7, IAL. Jan 2017) Core Maths 3 Functions Page 43