Inverse Functions

• The Concept of Reversibility: An inverse function $f^{-1}(x)$ is defined such that if the original function $f$ maps an input $x$ to an output $y$, then the inverse function maps $y$ back to $x$. This relationship is expressed as $f(x) = y \iff f^{-1}(y) = x$.

• Notation and Meaning: The notation $f^{-1}(x)$ refers specifically to the inverse function and should never be confused with the reciprocal $[f(x)]^{-1} = \frac{1}{f(x)}$. The superscript $-1$ indicates a functional inverse, not an algebraic exponent.

• The Identity Property: When a function is composed with its inverse, they cancel each other out to return the original input. This is written as $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$ for all values in the respective domains.

• The One-to-One Requirement: A function must be one-to-one (injective) to have an inverse. If a function is many-to-one, reversing the mapping would result in one input having multiple outputs, which fails the definition of a function.

• Horizontal Line Test: This is a visual tool used to determine if an inverse exists. If any horizontal line intersects the graph of $f(x)$ more than once, the function is many-to-one and does not have an inverse unless its domain is restricted.

• Symmetry and Reflection: Geometrically, the graph of $f^{-1}(x)$ is a reflection of the graph of $f(x)$ in the line $y = x$. This occurs because the inverse operation swaps the $x$ and $y$ coordinates of every point on the original curve.

Finding an Inverse Algebraically

• Step 1: Substitution: Replace the function notation $f(x)$ with the variable $y$ to make the equation easier to manipulate (e.g., $y = 2x + 3$).
• Step 2: Rearrangement: Solve the equation for $x$ in terms of $y$. This process isolates the input variable, effectively finding the 'reverse' rule.
• Step 3: Variable Swap: Interchange $x$ and $y$ so that the new equation is in the standard form $y = ...$. This step aligns the inverse with standard function notation.
• Step 4: Final Notation: Replace the new $y$ with $f^{-1}(x)$ to formally define the inverse function.

Decision Criteria for Domain Restrictions

• If a function is not one-to-one (like $f(x) = x^2$), you must restrict the domain (e.g., $x \ge 0$) before an inverse can be found. The choice of restriction determines which 'branch' of the function is being inverted.

• Inverse vs. Reciprocal: It is vital to distinguish between $f^{-1}(x)$ (the inverse function) and $\frac{1}{f(x)}$ (the multiplicative inverse). They are entirely different concepts; for example, the inverse of $x+2$ is $x-2$, while the reciprocal is $\frac{1}{x+2}$.

• The Domain-Range Swap: Always remember that the Domain of $f^{-1}$ is the Range of $f$. In exams, you are often asked to state the domain of the inverse; calculate the range of the original function to find this answer.

• Verification by Composition: To check if your algebraic inverse is correct, calculate $f(f^{-1}(x))$. If the result simplifies to $x$, your inverse is likely correct.

• Sketching Accuracy: When sketching an inverse, plot a few key points $(a, b)$ from the original function and reflect them to $(b, a)$. Ensure the curves intersect only on the line $y = x$.

• Common Mark Loser: Students often forget to write the final answer in terms of $x$. If the question asks for $f^{-1}(x)$, ensure your final expression does not contain the variable $y$.