User interface language: English | Español

Date May 2017 Marks available 3 Reference code 17M.2.AHL.TZ1.H_12
Level Additional Higher Level Paper Paper 2 Time zone Time zone 1
Command term Sketch Question number H_12 Adapted from N/A

Question

Consider f ( x ) = 1 + ln ( x 2 1 )

The function f is defined by f ( x ) = 1 + ln ( x 2 1 ) ,   x D

The function g is defined by g ( x ) = 1 + ln ( x 2 1 ) ,   x ] 1 ,   [ .

Find the largest possible domain D for f to be a function.

[2]
a.

Sketch the graph of y = f ( x ) showing clearly the equations of asymptotes and the coordinates of any intercepts with the axes.

[3]
b.

Explain why f is an even function.

[1]
c.

Explain why the inverse function f 1 does not exist.

[1]
d.

Find the inverse function g 1 and state its domain.

[4]
e.

Find g ( x ) .

[3]
f.

Hence, show that there are no solutions to  g ( x ) = 0 ;

[2]
g.i.

Hence, show that there are no solutions to  ( g 1 ) ( x ) = 0 .

[2]
g.ii.

Markscheme

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

x 2 1 > 0      (M1)

x < 1 or x > 1      A1

[2 marks]

a.

M17/5/MATHL/HP2/ENG/TZ1/12.b/M

shape     A1

x = 1 and x = 1      A1

x -intercepts     A1

[3 marks]

b.

EITHER

f is symmetrical about the y -axis     R1

OR

f ( x ) = f ( x )      R1

[1 mark]

c.

EITHER

f is not one-to-one function     R1

OR

horizontal line cuts twice     R1

 

Note:     Accept any equivalent correct statement.

 

[1 mark]

d.

x = 1 + ln ( y 2 1 )      M1

e 2 x + 2 = y 2 1      M1

g 1 ( x ) = e 2 x + 2 + 1 ,   x R      A1A1

[4 marks]

e.

g ( x ) = 1 x 2 1 × 2 x 2 x 2 1      M1A1

g ( x ) = x x 2 1      A1

[3 marks]

f.

g ( x ) = x x 2 1 = 0 x = 0      M1

which is not in the domain of g (hence no solutions to g ( x ) = 0 )     R1

 

[2 marks]

g.i.

( g 1 ) ( x ) = e 2 x + 2 e 2 x + 2 + 1      M1

as e 2 x + 2 > 0 ( g 1 ) ( x ) > 0 so no solutions to ( g 1 ) ( x ) = 0      R1

 

Note:     Accept: equation e 2 x + 2 = 0 has no solutions.

 

[2 marks]

g.ii.

Examiners report

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

Syllabus sections

Topic 2—Functions » SL 2.2—Functions, notation domain, range and inverse as reflection
Show 83 related questions
Topic 5 —Calculus » SL 5.3—Differentiating polynomials, n E Z
Topic 2—Functions » SL 2.4—Key features of graphs, intersections using technology
Topic 5 —Calculus » SL 5.6—Differentiating polynomials n E Q. Chain, product and quotient rules
Topic 2—Functions
Topic 5 —Calculus

View options