MEI Core 3 Functions / Natural logarithms and exponentials Chapter Assessment 1. The functions f, g and h are defined as follows: f ( x) e x x g( x) x x0 h( x) 2 x 1 x Find each of the following functions, giving the domain and range of each. (i) fg(x) (ii) gh(x) (iii) hf(x) -1 -1 (iv) f (x) (v) h (x) x x3 is odd. 1 x2 (ii) Explain what this tells you about the graph of y f ( x) . 2. (i) Show algebraically that the function f ( x) 3. (i) Sketch the graph of y 2 x 1 . (ii) Hence, or otherwise, solve each of the following equations: (a) 2x 1 3 x (b) 2 x 1 3x 2 [9] [6] [3] [1] [2] [3] [3] 4. Solve the equations (i) 2e x 3e x 5 (ii) ln(2 x 1) ln x 2 giving your answers in exact form. [3] [3] x 1 for x 0. x The graph approaches the line y = 1 as x becomes very large. 5. The diagram below shows the graph y = f(x), where f ( x) 1 1 (i) (ii) (iii) (iv) (v) Write down the domain and range of f(x). Find the inverse function f-1(x). Write down the domain and range of f-1(x). Sketch the graph of y = f-1(x) for the domain you gave in (iii). What is the relationship between the graph of y = f(x) and the graph of y = f-1(x)? © MEI, 10/05/10 [2] [3] [2] [2] [1] 1/8 MEI C3 Functions Assessment solutions 6. The graph of a function y = f(x) is shown below. The graph has a local maximum at ( 1, 1) and a local minimum at (2, 2). y (-1, 1) x (2, -2) Sketch the graphs of: (i) y 3f (2 x) (ii) y 2f ( x 1) (iii) y f (2 x) 1 (iv) y f ( x) 1 (v) y f ( x) (vi) giving the coordinates of the turning points in each case. [3] [3] [3] [3] [3] 7. A function is defined by f ( x) 2cos 1 x 1 . (i) Write down the domain and range of this function. (ii) Find the value of f(-0.5). (iii) Find the inverse function f 1 ( x) . [2] [2] [3] 8. (i) Solve the inequality 3x 2 4 . [3] (ii) Write the inequality 2 < x < 7 in the form x a b . [2] Total 70 marks © MEI, 10/05/10 2/8 MEI C3 Functions Assessment solutions Functions / Natural logarithms and exponentials Solutions to Chapter Assessment 1. (i) fg( x ) f( x ) e x Domain is x 0 Range is fg( x ) 1 (ii) gh( x ) g(2 x 1) 2 x 1 Domain is x 21 Range is gh( x ) 0 . Since g(x) is defined as a function, only the positive square root is used, giving this range. [3] [3] (iii) hf( x ) h(e ) 2e 1 Domain is x Range is hf( x ) 1 x x [3] (iv) y e ln y x x f 1( x ) ln x Domain is x 0 Range is f 1( x ) [3] (v) y 2 x 1 y 1 2x y 1 x 2 h1( x ) x 1 2 Domain is x Range is h1( x ) [3] 2. (i) x x3 1 x2 ( x ) ( x )3 x x 3 ( x x 3 ) f( x ) f( x ) 1 ( x )2 1 x2 1 x2 f( x ) so the function f is odd. [3] (ii) The graph of y f( x ) has rotational symmetry of order 2 about the origin. [1] © MEI, 10/05/10 3/8 MEI C3 Functions Assessment solutions 3. (i) y 2 x 1 1 21 [2] (ii) (a) 2x 1 3 x The graph shows that there are two solutions. 2x 1 3 x (2 x 1) 3 x 3x 2 x 2 x 1 3 x 4 x 2 3 Solutions are x 2 3 3 21 and x 4 [3] (b) 2 x 1 3x 2 The graph shows that there is just one solution 2 x 1 3x 2 21 3 x 2 3 The solution is x = 3. [3] 4. (i) 2e x 3e x 5 Substituting y e x : 2 y 3 y 1 5 Multiplying through by y: 2 y 2 3 5 y 2y2 5 y 3 0 (2 y 1)( y 3) 0 y 21 or 3 e x 21 or 3 Since e x cannot be negative, e x 3 x ln 3 [3] © MEI, 10/05/10 4/8 MEI C3 Functions Assessment solutions (ii) ln(2 x 1) ln x 2 ln(2 x 1) ln x 2 2x 1 ln 2 x 2x 1 e2 x 2 x 1 xe 2 1 xe 2 2 x 1 x(e 2 2) x 1 e 2 2 [3] 5. (i) Domain is x 0 Range is f( x ) 1 [2] x 1 (ii) y x xy x 1 1 x xy 1 x(1 y ) x 1 1y f 1( x ) 1 1x [3] (iii) Domain is x 1 Range is f( x ) 0 [2] (iv) 1 1 [2] (v) They are reflections of each other in the line y = x. [1] © MEI, 10/05/10 5/8 MEI C3 Functions Assessment solutions 6. (i) y 3f(2 x ) The graph is stretched scale factor 3 in the y direction, and scale factor in the x direction. The turning points are (1, -6) and ( 21 , 3). 1 2 ( 21 , 3) (1, -6) [3] (ii) y 2f( x 1) The graph is translated 1 unit horizontally to the right, and stretched scale factor 2 in the y direction. The turning points are (3, -4) and (0, 2) (0, 2) (3, -4) [3] (iii) y f(2 x ) 1 The graph is stretched scale factor 21 parallel to the x axis, and translated 1 unit downwards. The turning points are (1, -3) and ( 21 , 0) y ( 21 , 0) x (1, -3) [3] © MEI, 10/05/10 6/8 MEI C3 Functions Assessment solutions (iv) y f( x ) 1 The graph is reflected in the y-axis, and translated 1 unit upwards. The turning points are (-2, -1) and (1, 2). [3] (v) y f( x ) The negative parts of the graph are reflected in the x-axis. The turning points are (2, 2) and (-1, 1). [3] 1 7. f( x ) 2 cos x 1 (i) Domain for f(x) is 1 x 1 Range for cos 1 x is 0 x , so range for f(x) is 1 f( x ) 2 1 . [2] 1 (ii) f( x ) 2 cos ( 0.5 ) 1 2 2 1 3 4 1 3 [2] 1 (iii) y 2 cos x 1 y 1 2 cos 1 x y1 cos 1 x 2 y 1 2 x cos x 1 The inverse function is f 1( x ) cos 2 [3] © MEI, 10/05/10 7/8 MEI C3 Functions Assessment solutions 8. (i) 3x 2 4 4 3 x 2 4 2 3 x 6 23 x 2 (ii) 2 x 7 2 2.5 x 2.5 7 2.5 4.5 x 2.5 4.5 [3] x 2.5 4.5 [2] © MEI, 10/05/10 8/8

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# C3 Functions