Fundamental Common pitfalls

• Treating a formula as a full Function. A function also needs a Domain and codomain.
• Saying “the inverse of $x^2$ is $\sqrt{x}$” without restricting the domain. On $\mathbb{R}$, $x^2$ is not one-to-one; on $[0,\infty )$ it is invertible onto $[0,\infty )$.
• Confusing inverse notation with reciprocal notation. $f^{-1}(x)$ usually means inverse function, not $1/f(x)$.
• Ignoring composition order. $g\circ f$ means apply $f$ first, then $g$.
• Forgetting that $\ln x$ requires $x>0$ for real-valued logarithms.
• Cancelling terms across addition: $(a+b)/aeqb$ in general. Factor first, then cancel common non-zero factors.
• Dividing by a variable expression without checking whether it could be zero.
• Mixing exact and approximate values of Euler’s number. epprox2.71828 is useful numerically, but symbolic manipulation should keep $e$ exact where possible.
• Dropping units in applied problems. The derivative of position with respect to time is not just a number; it is Velocity with units such as m s${}^{-1}$.

Default rescue move: write down the domain, codomain, and restrictions before simplifying.