The main function families introduced so far form a common toolkit for algebra, geometry and calculus. Together they are often called elementary functions.

Main families

Important examples include polynomial, rational, power and root, exponential, logarithmic, trigonometric and inverse trigonometric functions.

Each family has characteristic domains and graph behavior. For example, polynomials are defined for every real input, rational functions may exclude denominator zeros, logarithms require positive inputs, and sine and cosine are periodic.

Building new functions

Elementary functions can be combined using arithmetic operations and composition. For example,

$$f(x)=e^{-x^2}$$

combines a polynomial expression, a negative sign and an exponential function, while

$$g(x)=\ln(1+x^2)$$

composes a logarithm with a polynomial expression.

The domain of a combined function must satisfy every restriction introduced by its parts.

Transformations of graphs

Starting from a known function $f$, expressions such as

$$f(x)+k,\qquad f(x-h),\qquad af(x),\qquad f(bx)$$

shift or scale its graph. Recognizing the underlying family makes it easier to anticipate domain, range, symmetry, periodicity and broad shape before calculating individual values.

This shared language of function families is the starting point for studying limits, continuity, derivatives and integrals.