Uploaded by Yae Hyun (Peter) LEE

Function I (2012)

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Function I (2012)
1. (a) (i) Show that x 2  4 x  7   x  22  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  1x 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
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