Function I (2012) 1. (a) (i) Show that x 2 4 x 7 x 22 a , where a is to be determined. (ii) Sketch the graph of y x 2 4 x 7 , giving the equation of the axis of symmetry and the coordinate of the vertex.. (b). The function f is defined by f : x x 2 4 x 7 and has its domain, the set of all real numbers. (i) Find the range of f. (ii) Explain, with reference to your sketch, why f has no inverse within the given domain. Suggest a domain for f for which it has an inverse, and find the inverse function in your suggested domain. 2. Given that f x 4kx 2 6x ( k 4) , find: (i). the values of k for which the equation f x 0 has real roots. (ii). The set of values of k for which f x is negative for all real values of x. 3. Find the set of values of k for which the equation f x 0 ,where f x k 1x 2 4 x k 2 Has (a) equal roots (b) distinct real roots (c) non-real roots. 4.. Find the set of values of k for which f x 4 x 2 4kx 3k 2 4k 3 is positive for all real values of x. Sketch the graph of y f x in the cases k 0 and k 1 . Explain how these graphs illustrate your result. 5. Show that the line y 3x 1 does not meet the curve y x 2 3x . 6. Find the condition that the line y mx c is a tangent to the circle x 2 y 2 4 x 0 . Hence find the equation of one of the tangents to the circle from the point 0 , 1 . 7. Find x in terms of y when y f x and hence find f 1 x when: (a) f x 7x 4 (b) f x 1 1 5x (c) f x 2x 5 . 7x 4 8. The functions f and g are defined with their respective domains by f : x 3 1 , x , x and 2x 1 2 g : x x 2 1, x (a) Find the values of x for which f x x . (b) Find the range of g. (c) The domain of the composite function f g is . Find f g x and state the range of f g. 9 All the following functions are mappings of , unless otherwise stated. (a) Determine the composite functions fg and gf , if they exist. (b) For the composite functions in (a), that do exist, find their range. (i) f x x 2 1 , g x (iii) f x 1 , x 0 (ii) f x 4 x , x 4 , g x x 2 x 1 2 , x 1 , g x x 1 , x 1 , g x x 2 1 (iv) f x x 1 x 1 10. Given that f x x 1, x and g f : x x 2 2 x 2 , x , determine the function g. 11. The functions f and g are defined by f : x x 1, x 1 x and g : x x , x \ 0 . Find the composite functions (where they exist) of (a) f g (b) g f (c) g g , stating the range in each case. 12. Given that f : x 2 x 1 , x , determine the two functions g, given that (a) g f x 1 2x 1 (b) f g x 1 . 2x 1