1.1 · Functions, domains, transformations, and inverses

A function assigns one output to each allowed input

The domain contains the inputs for which the formula is defined. For a real square root, require a nonnegative radicand; for a denominator, exclude zero; for a logarithm, require a positive argument.

The graph’s vertical-line test checks whether every input has at most one output. A graph may represent a function while failing the horizontal-line test for an inverse function.

Transform the input and output separately

In $\displaystyle g(x)=a f(b(x-h))+k$, the horizontal shift is $\displaystyle h$, the vertical shift is $\displaystyle k$, and nonzero $\displaystyle a,b$ control reflections and scaling. Horizontal distances scale by $\displaystyle 1/|b|$, while vertical distances scale by $\displaystyle |a|$.

For $\displaystyle g(x)=(x-2)^2+1$, the parent parabola moves right $\displaystyle 2$ and up $\displaystyle 1$. To read a composite horizontal transformation, factor the inner expression before identifying its shift.

$$g(x)=a f(b(x-h))+k$$

Composition and inversion must preserve domains

The composition $\displaystyle (f\circ g)(x)=f(g(x))$ requires both an allowed input for $\displaystyle g$ and an output of $\displaystyle g$ that belongs to the domain of $\displaystyle f$. Algebraic simplification does not restore excluded inputs.

To find an inverse, write $\displaystyle y=f(x)$, interchange input and output, and solve. Restrict the original domain when needed for one-to-one behavior. The inverse domain is the original range.