AP Calculus AB and BC glossary

Inverse function

An inverse function undoes another function: if g is the inverse of f, then f of g of x equals x and g of f of x equals x, so each undoes the other. A function has an inverse only when it is one-to-one. Arcsine, arctangent, and the natural log are defined this way.

Notation is $f^{-1}$ , and its graph is the reflection of $f$ across the line $y = x$ . Reflecting swaps coordinates, so the domain and range trade places: if $f$ sends $2$ to $7$ , then $f^{-1}$ sends $7$ back to $2$ .

An inverse exists only where $f$ is one-to-one, which the horizontal line test checks. Sine, cosine, and tangent fail that test on their full domains, so $\arcsin$ , $\arccos$ , and $\arctan$ are each built on a restricted piece where the function is one-to-one.

The mistake

Reading $f^{-1}(x)$ as $\frac{1}{f(x)}$ . The exponent notation here means the inverse function, not a reciprocal; $\sin^{-1} x$ is $\arcsin x$ , not $\frac{1}{\sin x}$ .

Appears in: Unit 3: Chain Rule, Implicit, and Inverses