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Answer to Question #5975 in Calculus for Samantha

Question #5975
Let f(x)=x/(x-6). Find a function y=g(x) so that (f*g)(x)=x.
Expert's answer
In this task we’re dealing with composition of functions. Recall the definition: function composition is the application of one function to the results of another. We need to find such function g(x) so that function f applied to it will give juxt x.

Due to the statement of the question f(x) = x/(x - 6)

To obtain function composition (f*g)(x)=f(g(x)) we have to substitute g(x) instead of x into the expression for
f(x):

as f(x) = x/(x - 6)

we have (f*g)(x) = f(g(x)) = g(x)/[g(x) - 6].

And the last expression has to be equal to x:

g(x)/[g(x) - 6]=x

Therefore

g(x) = xg(x) - 6x, so (x - 1)g(x) = 6x, and g(x) = 6x/(x - 1). Thus, g(x) is the inverse of f(x).

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