1.	Let f(x) = 7 – 2x and g(x) = x + 3.
(a)	Find (g ° f)(x).
(2)
 
(b)	Write down g–1(x).
(1)
 
(c)	Find (f ° g–1)(5).
(2)
(Total 5 marks)

2.	The functions f (x) and g (x) are defined by f (x) = ex and g (x) = ln (1+ 2x).
(a)	Write down f −1(x).
(b)	(i)	Find ( f ◦ g) (x).
(ii)	Find ( f ◦ g)−1 (x).
(Total 6 marks)

3.	(a)	Given that (2x)2 + (2x) −12 can be written as (2x + a)(2x + b), where a, b Î , find the value of a and of b.
(b)	Hence find the exact solution of the equation (2x)2 + (2x) −12 = 0, and explain why there is only one solution.
(Total 6 marks)

4.	Let f(x) = . Part of the graph of f is shown below.
		
	There is a maximum point at A and a minimum point at B(3, –9).
 
(a)	Find the coordinates of A.
(8)
 
(b)	Write down the coordinates of
(i)	the image of B after reflection in the y-axis;
(ii)	the image of B after translation by the vector ;
(iii)	the image of B after reflection in the x-axis followed by a horizontal stretch with scale factor .
(6)
(Total 14 marks)

5.	Let f(x) = cos 2x and g(x) = 2x2 – 1.
(a)	Find .
(2)
 
(b)	Find (g ° f).
(2)
 
(c)	Given that (g ° f)(x) can be written as cos (kx), find the value of k, k  .
(3)
(Total 7 marks)

6.	Let f(x) = x2 + 4 and g(x) = x – 1.
(a)	Find (f ° g)(x).
(2)
 
	The vector  translates the graph of (f ° g) to the graph of h.
(b)	Find the coordinates of the vertex of the graph of h.
(3)
 
(c)	Show that h(x) = x2 – 8x + 19.
(2)
 
(d)	The line y = 2x – 6 is a tangent to the graph of h at the point P. Find the x-coordinate of P.
(5)
(Total 12 marks)

7.	Let f(x) = ax2 + bx + c where a, b and c are rational numbers.
(a)	The point P(–4, 3) lies on the curve of f. Show that 16a –4b + c = 3.
(2)
 
(b)	The points Q(6, 3) and R(–2, –1) also lie on the curve of f. Write down two other linear equations in a, b and c.
(2)
 
(c)	These three equations may be written as a matrix equation in the form AX = B,
where X = .
(i)	Write down the matrices A and B.
(ii)	Write down A–1.
(iii)	Hence or otherwise, find f(x).
(8)
 
(d)	Write f(x) in the form f(x) = a(x – h)2 + k, where a, h and k are rational numbers.
(3)
(Total 15 marks)

8.	The diagram below shows the graph of a function f(x), for –2 ≤ x ≤ 4.

(a)	Let h(x) = f(–x). Sketch the graph of h on the grid below.
		
(2)
 
(b)	Let g(x) = f(x – 1). The point A(3, 2) on the graph of f is transformed to the point P on the graph of g. Find the coordinates of P.
(3)
(Total 5 marks)

9.	Part of the graph of a function f is shown in the diagram below.

(a)	On the same diagram sketch the graph of y = − f (x).
(2)
(b)	Let g (x) = f (x + 3).
(i)	Find g (−3).
(ii)	Describe fully the transformation that maps the graph of f to the graph of g.
(4)
(Total 6 marks)

10.	Let f be the function given by f(x) = e0.5x, 0 ≤ x ≤ 3.5. The diagram shows the graph of f.

(a)	On the same diagram, sketch the graph of f–1.
(3)
 
(b)	Write down the range of f–1.
(1)
 
(c)	Find f–1(x).
(3)
(Total 7 marks)