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Given that f(x) = 2x and g(x) = 3x^{2} find the values of x for which f^{-1}(x) = g(x)

(a) x=0 or 1/6

(b) x=0 or 2/3

(c) x=2/3 or 1/6

(d) x=1/6 or 2/3

**Comments**

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f(x) = 2x ------------- eqn(1)

g(x)=3x^{2} ------------- eqn(2)

to get the inverse of a function, f^{-1}(x), there are three steps:

1) replace f(x) with x and replace x with y

from eqn(1),

x = 2y

2) make y the subject of the formula

y = x/2

3) replace y with f^{-1}(x)

f^{-1}(x) = x/2

Thereofre, f^{-1}(x) = g(x) will be:

x/2 = 3x^{2}

cross multiply

x = 6x^{2}

6x^{2} - x = 0

x(6x - 1) = 0

x = 0 and 6x - 1 = 0

x = 0 and x = 1/6

**Answer A**

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