Squared and Reciprocal R-form

Let $t=3\sin\theta+4\cos\theta$. Over a full turn, $-5\le t\le5$. Squaring does not preserve that order: both endpoints give 25, while $t=0$ gives the minimum 0.

For a shifted square such as $(2+t)^2$, first find $-3\le2+t\le7$. This interval contains zero, so the minimum square is 0; the larger endpoint magnitude is 7, so the maximum is 49.

For $1/(6+t)$, the denominator ranges from 1 to 11. Both are positive: the larger denominator gives the smaller reciprocal.

For $1/t$, zero is excluded and values near zero produce arbitrarily large positive or negative outputs. The two parts of $-5\le t\le5$ give $y\le-1/5$ or $y\ge1/5$, not a single interval between those numbers.