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

Yes — one successful numeric test is sufficient

No — numeric tests can screen for non-inverses but cannot prove the relationship for all $x$; symbolic verification is needed

Yes — if it works at $x = 4$, it works for all $x$

No — the student should test $f(g(x))$ symbolically and find that it does not simplify to $x$

No — they look like inverses but aren't

Yes — $f(p(x)) = (x-5)+5 = x$ and $p(f(x)) = (x+5)-5 = x$

No — only $f(p(x)) = x$; $p(f(x)) \neq x$

Cannot be determined without a graph

Both equal $x$ — they are inverses

$f(q(x)) = 2x + 6$ and $q(f(x)) = 2x + 3$; neither equals $x$, so they are not inverses

They are equal: $f(q(x)) = q(f(x)) = 2x + 3$

The functions cannot be composed

The student computed the composition incorrectly

The student checked only one direction — $g(f(x)) = x$ must also be verified

One direction is always sufficient to conclude inverse relationship

Symbolic verification is not valid — only numeric checks are valid

The algebra in the verification step is wrong

The student only computed one direction — $f(g(x))$ — and declared both directions verified without computing $g(f(x))$

The inverse function $g(x)$ is incorrect

The computation is complete — one direction suffices when the algebra is clean

$f(x) = x^3$ and $g(x) = x^{1/3}$

$f(x) = e^x$ and $g(x) = \ln(x)$

$f(x) = 2x + 1$ and $g(x) = 2x - 1$

$f(x) = \frac{x+5}{3}$ and $g(x) = 3x - 5$