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Date November 2021 Marks available 1 Reference code 21N.1.AHL.TZ0.2
Level Additional Higher Level Paper Paper 1 (without calculator) Time zone Time zone 0
Command term Sketch Question number 2 Adapted from N/A

Question

The function f is defined by fx=2x+43-x, where x, x3.

Write down the equation of

Find the coordinates where the graph of f crosses

the vertical asymptote of the graph of f.

[1]
a.i.

the horizontal asymptote of the graph of f.

[1]
a.ii.

the x-axis.

[1]
b.i.

the y-axis.

[1]
b.ii.

Sketch the graph of f on the axes below.

[1]
c.

The function g is defined by gx=ax+43-x, where x, x3 and a.

Given that gx=g-1x, determine the value of a.

[4]
d.

Markscheme

x=3                 A1

 

[1 mark]

a.i.

y=-2                 A1

 

[1 mark]

a.ii.

-2,0   (accept x=-2)                 A1

 

[1 mark]

b.i.

0,43   (accept y=43 and f0=43)                 A1

 

[1 mark]

b.ii.

               A1


Note:
Award A1 for completely correct shape: two branches in correct quadrants with asymptotic behaviour.

 

[1 mark]

c.

METHOD 1

gx=y=ax+43-x

attempt to find x in terms of y                (M1)

OR exchange x and y and attempt to find y in terms of x

3y-xy=ax+4                A1

ax+xy=3y-4

xa+y=3y-4

x=3y-4y+a

g-1x=3x-4x+a                A1


Note: Condone use of y=


gxg-1x

ax+43-x3x-4x+a

a=-3                A1

 

METHOD 2

gx=ax+43-x

attempt to find an expression for ggx and equate to x                (M1)

ggx=aax+43-x+43-ax+43-x=x                A1

aax+4+43-x9-3x-ax+4=x

aax+4+43-x5-3+ax=x

aax+4+43-x=x5-3+ax                A1

equating coefficients of x2  (or similar)

a=-3                A1

 

[4 marks]

d.

Examiners report

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

Syllabus sections

Topic 2—Functions » SL 2.2—Functions, notation domain, range and inverse as reflection
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Topic 2—Functions » SL 2.5—Composite functions, identity, finding inverse
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