The function f is defined by

1.
f (x) = (x2 + 1), x ≥ 0

(a) Find the range of f.

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(1)

(b) (i) Find f −1(x)

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(ii) State the range of f −1(x)

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(1)

(c) State the transformation which maps the graph of y = f (x) onto the graph of
y = f −1(x)

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(1)

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 (d) Find the coordinates of the point of intersection of the graphs of y = f (x) and
y = f −1(x)

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(2)
(Total 8 marks)

The function f is defined by

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f (x) = 4 + 3−x , x ∈ ℝ

(a) Using set notation, state the range of f

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(2)

(b) The inverse of f is f−1

(i) Using set notation, state the domain of f−1

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(1)

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 (ii) Find an expression for f−1(x)

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(3)

(c) The function g is defined by

, (x ∈ ℝ : x > 0)

(i) Find an expression for gf(x)

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(1)

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 (ii) Solve the equation gf(x) = 2, giving your answer in an exact form.

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(3)
(Total 10 marks)

The function f is defined by

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f(x) = , for real values of x, where x ≤ 0

(a) State the range of f.

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(2)

(b) The inverse of f is f −1.

(i) Write down the domain of f −1.

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(1)

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 (ii) Find an expression for f −1(x).

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(3)

(c) The function g is defined by

g(x) = ln |3x − 1|, for real values of x, where x ≠

The curve with equation y = g(x) is sketched below.

(i) The curve y = g(x) intersects the x-axis at the origin and at the point P.

Find the x-coordinate of P.

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(2)

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(ii) State whether the function g has an inverse. Give a reason for your answer.

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(1)

(iii) Show that gf(x) = ln |x2 − k|, stating the value of the constant k.

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(2)

(iv) Solve the equation gf(x)= 0.

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(4)
(Total 15 marks)

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 The curve with equation y = f(x), where f(x) = ln(2x − 3), x > , is sketched below.
4.

(a) The inverse of f is f −1.

(i) Find f −1(x).

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(3)

(ii) State the range of f −1.

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(1)

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 (iii) Sketch, on the axes given on the opposite page, the curve with equation y = f −1(x),
indicating the value of the y-coordinate of the point where the curve intersects the
y-axis.

(2)

(b) The function g is defined by

g(x)= e2x − 4, for all real values of x

(i) Find gf(x), giving your answer in the form (ax − b)2 − c, where a, b and c are integers.

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(3)

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 (ii) Write down an expression for fg(x), and hence find the exact solution of the equation
fg(x) = ln 5.

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(3)
(Total 12 marks)

The polynomial f(x) is defined by f(x) = 2x3 + x2 − 8x − 7.

5.
(a) Find the remainder when f(x) is divided by (2x + 1).

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(2)

(b) The polynomial g(x) is defined by g(x) = f(x) + d, where d is a constant.

(i) Given that (2x + 1) is a factor of g(x), show that g(x) = 2x3 + x2 − 8x − 4.

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(1)

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(ii) Given that g(x) can be written as g(x) = (2x + 1)(x2 + a), where a is an integer,
express g(x) as a product of three linear factors.

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(1)

(iii) Hence, or otherwise, show that , where p and q are integers.

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(3)
(Total 7 marks)

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 (a) Sketch the graph of
6.
y = 4 − |2x − 6|

(3)

(b) Solve the inequality

4 − |2x − 6| > 2

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(2)
(Total 5 marks)

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7. The diagram shows the curves y = e2x – 1 and y = 4e–2x + 2.

The curve y = 4e–2x + 2 crosses the y-axis at the point A and the curves intersect at the point B.

(a) Describe a sequence of two geometrical transformations that maps the graph of y = ex onto
the graph of y = e2x – 1.

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(4)

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(b) Write down the coordinates of the point A.

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(1)

(c) (i) Show that the x-coordinate of the point B satisfies the equation

(e2x)2 – 3e2x – 4 = 0

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(ii) Hence find the exact value of the x-coordinate of the point B.

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(3)

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(d) Find the exact value of the area of the shaded region bounded by the curves y = e2x – 1
and y = 4e–2x + 2 and the y-axis.

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(5)
(Total 15 marks)

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8. Find the value of expressing your answer in the form mln2 + nln3 , where
m and n are integers.

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(Total 8 marks)

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 (a) The diagram shows part of the curve with equation y = f(x). The curve crosses the x-axis at
9.
the point (, 0) and the y-axis at the point (0, –b).

On a separate diagram, sketch the curve with the equation y = 2f(x). On the diagram,
indicate, in terms of a or b, the coordinates of the points where the curve crosses the
coordinate axes.

(2)

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(b) (i) Describe a sequence of geometrical transformations that maps the graph of y = ln x
onto the graph of y = 4 ln(x + 1) – 2.

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(6)

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 (ii) Find the exact values of the coordinates of the points where the graph of
y = 4 ln(x + 1) – 2 crosses the coordinate axes.

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(4)
(Total 12 marks)

The graph of y = f(x) is shown below.

10.

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(a) Sketch the graph of y = f(−x)

(2)

(b) Sketch the graph of y = 2f(x) − 4

(2)

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(c) Sketch the graph of y = f ʹ(x)

(3)
(Total 7 marks)

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 The diagram below shows the graphs of y = |2x − 3| and y = |x|.
11.

(a) Find the x-coordinates of the points of intersection of the graphs of y = |2x − 3| and y = |x|.

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(3)

(b) Hence, or otherwise, solve the inequality

|2x − 3| ≥ |x|

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(2)
(Total 5 marks)

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 (a) On separate diagrams:
12.
(i) sketch the curve with equation y =│3x + 3│;

(2)

(ii) sketch the curve with equation y =│x2 – 1│.

(3)

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(b) (i) Solve the equation│3x + 3│=│x2 – 1│.

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(5)

(ii) Hence solve the inequality │3x + 3│<│x2 – 1│.

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(2)
(Total 12 marks)

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