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Date May 2018 Marks available 3 Reference code 18M.1.AHL.TZ2.H_10
Level Additional Higher Level Paper Paper 1 Time zone Time zone 2
Command term Sketch and State Question number H_10 Adapted from N/A

Question

The function  f is defined by  f ( x ) = a x + b c x + d , for  x R , x d c .

The function  g is defined by  g ( x ) = 2 x 3 x 2 , x R , x 2

Find the inverse function  f 1 , stating its domain.

[5]
a.

Express  g ( x ) in the form  A + B x 2  where A, B are constants.

[2]
b.i.

Sketch the graph of  y = g ( x ) . State the equations of any asymptotes and the coordinates of any intercepts with the axes.

[3]
b.ii.

The function  h  is defined by  h ( x ) = x , for  x  ≥ 0.

State the domain and range of  h g .

[4]
c.

Markscheme

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

attempt to make x the subject of  y = a x + b c x + d       M1

y ( c x + d ) = a x + b       A1

x = d y b a c y      A1

f 1 ( x ) = d x b a c x      A1

Note: Do not allow  y = in place of f 1 ( x ) .

x a c , ( x R )      A1

Note: The final A mark is independent.

[5 marks]

a.

g ( x ) = 2 + 1 x 2      A1A1

[2 marks]

b.i.

hyperbola shape, with single curves in second and fourth quadrants and third quadrant blank, including vertical asymptote  x = 2      A1

horizontal asymptote  y = 2      A1

intercepts  ( 3 2 , 0 ) , ( 0 , 3 2 )      A1

[3 marks]

b.ii.

the domain of  h g is  x 3 2 , x > 2      A1A1

the range of  h g is  y 0 , y 2      A1A1

[4 marks]

c.

Examiners report

[N/A]
a.
[N/A]
b.i.
[N/A]
b.ii.
[N/A]
c.

Syllabus sections

Topic 2—Functions » SL 2.2—Functions, notation domain, range and inverse as reflection
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