Exercises: (+) Verify by Composition That One Function Is the Inverse of Another

Only $f(g(x)) = x$ must hold

Only $g(f(x)) = x$ must hold

Both $f(g(x)) = x$ AND $g(f(x)) = x$ must hold

$f(g(x)) = g(f(x))$ must hold (they must be equal to each other, not necessarily to $x$)

One check is always sufficient for simple functions

There exist function pairs where $f(g(x)) = x$ but $g(f(x)) \neq x$

The second check always gives the same result as the first

Numeric checks are enough — symbolic checks are not required

Yes

No — $f(g(x)) = x + 3$

No — $f(g(x)) = 5x - 3$

No — $f(g(x)) = x - 3$

Yes

No — $g(f(x)) = 5x - 12$

No — $g(f(x)) = x + 3$

No — $g(f(x)) = x - 3$

Yes — $(\sqrt{x})^2 = x$ always

No — $f(g(x)) = x$ only for $x \geq 0$; $g(x) = \sqrt{x}$ is undefined for $x < 0$

No — the composition gives $|x|$, not $x$

Yes — both $f(g(x)) = x$ and $g(f(x)) = x$ hold everywhere