For which pairs of functions is ?

Asked by maham237 @ 25/12/2020 in Mathematics viewed by 293 People

For which pairs of functions is f(g(x))=x?
A) f(x)=x^2 and g(x)=1/x

B) f(x)=2/x and g(x)=2/x

C) f(x)x-2/3 and g(x)=2-3x

D) f(x)1/2x-2 and g(x)=1/2x+2

Answered by maham237 @ 25/12/2020

The right answer is B) f(x)=2/x and g(x)=2/x

This is a problem of composition of functions.

$The \ \mathbf{composition} \ of \ the \ function \ f \ with \ the \ function \ g \ is:\\ \\ (f \circ g)(x)=f(g(x)) \\ \\ The \ domain \ of \ (f \circ g) \ is \ the \ set \ of \ all \ x \ in \ the \ domain \ of \ g \\ such \ that \ g(x) \ is \ in \ the \ domain \ of \ f$

So, the only functions that satisfies this concept is B). To prove this, let's take the following steps:

$f(x)=\frac{2}{x} \ and \ g(x)=\frac{2}{x} \\ \\ f(g(x))=\frac{2}{\frac{2}{x}}=\frac{2x}{2} \\ \\ \therefore \boxed{f(g(x))=x}$

Since this composition of functions gives us the identity function, then we can conclude that the function g is the inverse of f and vice versa.

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