Date | May 2021 | Marks available | 1 | Reference code | 21M.2.AHL.TZ2.12 |
Level | Additional Higher Level | Paper | Paper 2 | Time zone | Time zone 2 |
Command term | Show that | Question number | 12 | Adapted from | N/A |
Question
A function f is defined by f(x)=arcsin(x2-1x2+1), x∈ℝ.
A function g is defined by g(x)=arcsin(x2-1x2+1), x∈ℝ, x≥0.
Show that f is an even function.
By considering limits, show that the graph of y=f(x) has a horizontal asymptote and state its equation.
Show that f'(x)=2x√x2(x2+1) for x∈ℝ, x≠0.
By using the expression for f'(x) and the result √x2=|x|, show that f is decreasing for x<0.
Find an expression for g-1(x), justifying your answer.
State the domain of g-1.
Sketch the graph of y=g-1(x), clearly indicating any asymptotes with their equations and stating the values of any axes intercepts.
Markscheme
EITHER
f(-x)=arcsin((-x)2-1(-x)2+1)=arcsin(x2-1x2+1)=f(x) R1
OR
a sketch graph of y=f(x) with line symmetry in the y-axis indicated R1
THEN
so f(x) is an even function. AG
[1 mark]
as x→±∞, f(x)→arcsin 1(→π2) A1
so the horizontal asymptote is A1
[2 marks]
attempting to use the quotient rule to find M1
A1
attempting to use the chain rule to find M1
let and so and
M1
A1
A1
AG
[6 marks]
EITHER
for (A1)
so A1
OR
and A1
A1
THEN
R1
Note: Award R1 for stating that in , the numerator is negative, and the denominator is positive.
so is decreasing for AG
Note: Do not accept a graphical solution
[3 marks]
M1
A1
A1
domain of is and so the range of must be
hence the positive root is taken (or the negative root is rejected) R1
Note: The R1 is dependent on the above A1.
so A1
Note: The final A1 is not dependent on R1 mark.
[5 marks]
domain is A1
Note: Accept correct alternative notations, for example, or .
Accept if correct to s.f.
[1 mark]
A1A1A1
Note: A1 for correct domain and correct range and -intercept at
A1 for asymptotic behaviour
A1 for
Coordinates are not required.
Do not accept or other inexact values.
[3 marks]