Mathematics · May / June 2021 · Paper 31 · Question 4
YMA01
As printed in the paper
The question as text
4. The functions f and g are defined by f (x) = 4 x + 6 x , x ¹ 5 x − 5 g (x) = 5 − 2x2 x , x 0
Read automatically from the paper; the printed page above is exact.
Show the answer and mark scheme
4(a)[4]
4 5
−
2
x
2
+
6
fg( )
x
=
2
5
−
2
x
−
5
26 8
−
x
2
fg( )
x
=
3
⇒
=
3
⇒
x
= −
13
−
2
x
2
4(b)[3]
4
x
+
6
y
=
⇒
yx
−
5
y
=
4
x
+
6
⇒
y x
−
4
x
=
5
y
+
6
x
−
5
5
y
+
6
⇒
x
=
y
−
4
5
x
+
6
⇒
f
−
1
( )
x
=
x
∈
,
x
≠
4
x
−
4
4(c)[3]
Shape and position for y = g(x). A
parabola starting on the yaxis that
extends down in quadrant 2 to at least
the xaxis. Condone slips of the pen
e.g. if the curve heads back towards
the yaxis in quadrant 3 slightly.
Shape and position for y = g(x) as
above that crosses the xaxis and
5
y = g1(x) a parabola starting on the x
axis with graphs crossing as shown in
y
=
g( )
x
quadrant 3. Condone slips of the pen
e.g. if g1(x) heads back towards the x
5
axis in quadrant 3 slightly.
5
−
O
2
Correct intercepts. This mark is for
y
=
g ( )
−
1
x
5
−
the 4 correct intercepts as shown.
2
Allow as values or coordinates and
allow coordinates the wrong way
round (e.g. (0, 5) for (5, 0)) as long as
they are in the correct places. The
negative intercepts must be exact as
shown or equivalent. The graphs must
cross the axes at these points and you
can ignore any other intercepts.
Note that labels for g(x) and g1(x) are not required
(a)
M1: For attempting fg( )
x in the correct order.
dM1: For proceeding to
Depends on the first mark.
x =
2
...
A1: For
x =
2
13
which may be implied by
x =
13
A1: For
13
only
x = −
1
2
( )
−
( )
fg( )
x
=
3
⇒
g
x
=
f
3
= −
21
⇒
5
−
2
x
= −
21
Score M1 for finding f −1(3) and setting g(x) = f −1(3) then as main scheme
or
( )
−
1
( )
−
1
(
)
fg( )
x
=
3
⇒
g
x
=
f
3
= −
21
⇒
x
=
g
−
21
Score M1 for finding f −1(3) and setting g−1f −1(3) = x then as main scheme
(b)
M1: For attempting to change the subject. In the attempt shown in the mark scheme score when both terms
in x have been isolated and set on one side of the equation
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