Common Mistakes

Frequent Errors

1. Writing $f^{-1}(x)=1/f(x)$ instead of the inverse function. 2. Claiming $f(x)=x^2$ or $|x|$ is invertible on all reals. 3. Forgetting to restrict domain before finding an inverse. 4. Swapping $x$ and $y$ but solving incorrectly. 5. Leaving Dom$(f^{-1})$ as Dom$(f)$ instead of Range$(f)$. 6. Reflecting over the wrong line (e.g., $x$- or $y$-axis). 7. Skipping the identity composition check. 8. Ignoring contextual restrictions (2.8.B.5).

Why These Happen

The $-1$ exponent notation looks like a reciprocal, and students often skip one-to-one checks.

How to Fix Them

• Say "inverse undoes; reciprocal multiplies to 1."
• Horizontal line test on every candidate.
• Always state the invertible domain before writing $f^{-1}$.
• Verify with $f(f^{-1}(x))=x$.

Avoidance Scripts

• "One output, one input — then invertible."
• "Swap pairs; swap axes; swap and solve."
• "Domain of inverse = range of original."

Quick Self-Check

• Is $f$ one-to-one on this domain?
• Did I swap pairs/domain/range correctly?
• Does composition return $x$?