Date | November 2020 | Marks available | 3 | Reference code | 20N.2.SL.TZ0.S_4 |
Level | Standard Level | Paper | Paper 2 | Time zone | Time zone 0 |
Command term | Find and Hence or otherwise | Question number | S_4 | Adapted from | N/A |
Question
Let f(x)=4-x3 and g(x)=ln x, for x>0.
Find (f∘g)(x).
Solve the equation (f∘g)(x)=x.
Hence or otherwise, given that g(2a)=f-1(2a), find the value of a.
Markscheme
* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.
attempt to form composite (in any order) (M1)
eg f(ln x) , g(4-x3)
(f∘g)(x)=4-(ln x)3 A1 N2
[2 marks]
valid approach using GDC (M1)
eg , (2.85, 2.85)
2.85056
2.85 A1 N2
[2 marks]
METHOD 1 – (using properties of functions)
recognizing inverse relationship (M1)
eg f(g(2a))=f(f-1(2a)) (=2a)
equating 2a to their x from (i) (A1)
eg 2a=2.85056
1.42528
a=1.43 A1 N2
METHOD 2 – (finding inverse)
interchanging x and y (seen anywhere) (M1)
eg x=4-y3 , f-1(x)=3√4-x
correct working (A1)
eg 3√4-2a=ln(2a), sketch showing intersection of f-1(2x) and g(2x)
1.42528
a=1.43 A1 N2
[3 marks]