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1 C3 Functions. June 200 qu.9 The functions f and g are defined for all real values of b f() = and g() = a + b, where a and b are non-zero constants. (i) Find the range of f. [3] Eplain wh the function f has no inverse. (iii) Given that g () = g() for all values of, show that a =. [4] (iv) Given further that gf() < 5 for all values of, find the set of possible values of b. [4] 2. Jan 200 qu.4 3 The function f is defined for all real values of b f() = 2 +. The diagram shows the graph of = f(). (i) Evaluate ff( 26). Find the set of values of for which f() = f(). (iii) Find an epression for f (). [3] (iv) State how the graphs of = f() and = f () are related geometricall. [] 3. June 2009 qu.5 The functions f and g are defined for all real values of b f() = 3 2 and g() = Find the eact coordinates of the point at which (i) the graph of = fg() meets the -ais, [3] the graph of = g() meets the graph of = g (), [3] (iii) the graph of = f() meets the graph of = g(). [4] 4. June 2009 qu.8 The diagram shows the curves = ln and = 2 ln( 6). The curves meet at the point P which has -coordinate a. The shaded region is bounded b the curve = 2 ln( 6) and the lines = a and = 0. (i) Give details of the pair of transformations which transforms the curve = ln to the curve = 2 ln( 6). [3] Solve an equation to find the value of a. [4]

2 5. Jan 2009 qu.6 The function f is defined for all real values of b f() = The graphs of = f() and = f () meet at the point P, and the graph of = f () meets the -ais at Q (see diagram). (i) Find an epression for f () and determine the -coordinate of the point Q. [3] State how the graphs of = f() and = f () are related geometricall, and hence show that the -coordinate of the point P is the root of the equation = Jan 2009 qu.7 The diagram shows the curve = e k a, where k and a are constants. (i) (iii) Give details of the pair of transformations which transforms the curve = e to the curve = e k a. [3] Sketch the curve = e k a. Given that the curve = e k a passes through the points (0, 3) and (ln 3, 3), find the values of k and a. [4] 7. June 2008 qu. Find the eact solutions of the equation 4 5 = 3 5. [4]

3 8. June 2008 qu.2 The diagram shows the graph of = f(). It is given that f( 3) = 0 and f(0) = 2. Sketch, on separate diagrams, the following graphs, indicating in each case the coordinates of the points where the graph crosses the aes: (i) = f (), = 2f(). [3] 9. June 2008 qu.7 It is claimed that the number of plants of a certain species in a particular localit is doubling ever 9 ears. The number of plants now is 42. The number of plants is treated as a continuous variable and is denoted b N. The number of ears from now is denoted b t. (i) Two equivalent epressions giving N in terms of t are N = A 2 kt and N = Ae mt. Determine the value of each of the constants A, k and m. [4] Find the value of t for which N = 00, giving our answer correct to 3 significant figures. (iii) Find the rate at which the number of plants will be increasing at a time 35 ears from now. [3] 0. Jan 2008 qu. Functions f and g are defined for all real values of b f() = and g() = 2 5. Evaluate (i) fg(), f (2). [3]. Jan 2008 qu.6 The diagram shows the graph of = sin ( ). (i) Give details of the pair of geometrical transformations which transforms the graph of = sin ( ) to the graph of = sin. [3] Sketch the graph of = sin ( ). ( ). (iii) Find the eact solutions of the equation sin = π 3 [3]

4 2. June 2007 qu.2 Solve the inequalit 4 3 < 2 +. [5] 3. June 2007 qu.3 The function f is defined for all non-negative values of b f() = 3 +. (i) Evaluate ff(69). Find an epression for f () in terms of. (iii) n a single diagram sketch the graphs of = f() and = f - (), indicating how the two graphs are related. [3] 4. June 2007 qu.5 A substance is decaing in such a wa that its mass, m kg, at a time t ears from now is given b the formula m = 240e 0.04t (i) Find the time taken for the substance to halve its mass. [3] Find the value of t for which the mass is decreasing at a rate of 2. kg per ear. [4] 5. Jan 2007 qu.9 Functions f and g are defined b f() = 2 sin for π π, g() = for. (i) State the range of f and the range of g. Show that gf (0. 5) = 2.6, correct to 3 significant figures, and eplain wh fg (0. 5) is not defined. [4] (iii) Find the set of values of for which f g() is not defined. [6] 6. June 2006 qu.2 Solve the inequalit 2 3 < +. [5] 7. June 2006 qu.6 The diagram shows the graph of = f(), where f() = 2 2, 0. (i) Evaluate ff( 3). [3] Find an epression for f (). [3] (iii) Sketch the graph of = f (). Indicate the coordinates of the points where the graph meets the aes. [3]

5 8. Jan 2006 qu.4 The function f is defined b f() = 2 for 0. The graph of = f() is shown above. (i) State the range of f. [] Find the value of ff(4). (iii) Given that the equation f() = k has two distinct roots, determine the possible values of the constant k. 9. June 2005 qu. The function f is defined for all real values of b f() = 0 ( + 3) 2. (i) State the range of f. [] Find the value of ff( ). [3] 20. June 2005 qu.2 Find the eact solutions of the equation 6 =. [4] 2. June 2005 qu.9 = f ( ) 7 m 4 7 The function f is defined b f () = (m + 7) 4, where and m is a positive constant. m The diagram shows the curve = f(). (i) A sequence of transformations maps the curve = to the curve = f(). Give details of these transformations. [4] Eplain how ou can tell that f is a one one function and find an epression for f (). [4] (iii) It is given that the curves = f() and = f () do not meet. Eplain how it can be deduced that neither curve meets the line =, and hence determine the set of possible values of m. [5]

and hence show that the only stationary point on the curve is the point for which x = 0. [4]

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