The diagram shows the graph of y = f^−1 (x), where f^ −1 is defined by f^−1 (x) = (1 − 5x)/2x for 0<x≤ 2.
(i) Find an expression for f(x) and state the domain of f. [5]
(ii) The function g is defined by g(x) =1/x for x ≥ 1. Find an expression for f^ −1 g(x), giving your
answer in the form ax + b, where a and b are constants to be found. [2]
Solutions:
i)
Explanation:
If x = 1
f^-1 (1) = 1-5(1) / 2(1)
= -4/2
= -2
If x = 2
f^-1(2) = 1-5(2) / 2(2)
= 1-10 / 4
= -9/4
= -2 1/4 (smaller than -2)
If x = 0.1
f^-1 (0.1) = 1-5(0.1) / 2(0.1)
= 2.5
If x = 0.01
f^-1 (0.01) = 1-5(0.01) / 2(0.01)
= 47.5
If x = 0.00001
f^-1 (0.00001) = 1-5(0.00001) / 2(0.00001)
= 49997.5
Actually the range of f^-1(x) is the domain of f(x). When I subs x = 2 to the f^-1(x), the range is -9/4 (this one no issue). When subs x = 1, the range become -2.
Why the domain not equal to -9/4 <= x <= -2, but x >= -9/4 only?
You will understand when you see my solutions for x = 0.1, x=0.01, x = 0.00001 ...
x can be 0.000000000000000000000000............1 (it also can be 0.05, 0.12345678 and any number which is more than 0 but less than 2) as long as it is bigger than 0. So the range will be any number which is more than or equal to -9/4
ii)
Reference: PYQ - May/June 2015 Paper 13 Q6
