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    4 kx 2  6 x ( k 4) , find:
(i). the values of k for which the equation    0 has real roots.
(ii). The set of values of k for which is negative for all real values of x.
3. Find the set of values of k for which the equation f
 

0
,where f
   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

4 x
2 
4 kx


3 k
2 
4 k

3

is positive for all real values of x. Sketch the graph of y
 f
  in the cases k

0 and k

1
. Explain how these graphs illustrate your result.
5. Show that the line y

3 x

1 does not meet the curve y
 x
2 
3 x
.
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    and hence find f  1
  when:
(a)    7 x  4 (b) 
1 x
(c) 
2 x  5
7 x  4
.

8. The functions f and g are defined with their respective domains by 
3
2 x

1
, x

, x

1
2
and
:
 x
2 
1, x

(a) Find the values of x for which f
 
 x
.
(b) Find the range of g.
(c) The domain of the composite function f  g is . Find f  g
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)    x 2 
1 ,

1 x
, x

0
(ii) 
4

,

4 ,
   x 2
(iii) 
2 x

1
, x

1 ,
   x
2 
1
(iv) 
1 x

1
, x
 
1 ,
  x 1
10. Given that   x 1, x
 and g :
 x
2 
2 x

2 , x
 , determine the function g.
11. The functions f and g are defined by
:
 
1, x
 and
:
 
1 x
, x
 .
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
:

2 x

1 , x
 , determine the two functions g, given that
(a) g
  
1
2 x

1
(b) f
  
1
2 x

1
.