Consider the functions f and g where f(x)=3x-5 and g(x)=x-2. (a) Find the inverse function for f. (b) Given that the inverse of g is x+2, find (g-1 o f)(x).
(a) In order to find the inverse of a function, it is easiest to swap x and y and solve for y. Here this would give, x=3y-5 => x+5=3y => (x+5)/3=y. Hence, f-1(x)=(x+5)/3. (b) Here it is important to remember the order in which to calculate the composition of a function and then slowly plugging in the required functions. This gives (g-1 o f)(x) = g-1(f(x))= g-1(3x-5)=3x-5+2=3x-3.
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