How do you show that $f (x) = \frac{1}{x}$ $f(x)=1/x$ and $g (x) = \frac{1}{x}$ $g(x)=1/x$ are inverse functions algebraically and graphically?

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Answer 1 · 2018-02-17T21:35:36

See below.

If $f (x)$ $f(x)$ and $g (x)$ $g(x)$ are inverse functions, then:

$f (g (x)) = g (f (x)) = x$ $f(g(x))=g(f(x))=x$

$\frac{1}{g (x)} = \frac{1}{f (x)} = x$ $1/g(x)=1/f(x)=x$

$\frac{1}{\frac{1}{x}} = \frac{1}{\frac{1}{x}} = x$ $1/(1/x)=1/(1/x)=x$

$x = x = x$ $x=x=x$

Since these are there own inverses we would expect their graphs to be symmetrical about the line $y = x$ $y=x$

The graph below confirms this:

enter image source here

How do you show that f(x)=1xf(x)=1/x and g(x)=1xg(x)=1/x are inverse functions algebraically and graphically?