Some Special Functions

Named function families.

• Polynomial function. $f:R\to R,\ f(x)=a_0x^n+a_1x^{n-1}+\cdots+a_{n-1}x+a_n$ ($a_i$ constants) -- named because the right side is literally a polynomial.
• Linear function. $f(x)=ax+b$ ($a\ne0,\ b$ constants); its graph is a straight line, hence the name. (A function that isn't linear is called non-linear.) Every linear function is automatically a polynomial function (degree 1).
• Exponential function. For a constant $a>0$, $f(x)=a^x$. If $a=1$, this collapses to the constant function $f(x)=1$; if $a>1$ it is genuinely called exponential. (Any function with the variable in the exponent is loosely called exponential.) $e$ is a special irrational constant between $2$ and $3$, central to later chapters.
• Logarithmic function. For a constant $a>1$, $f:(0,\infty)\to R,\ f(x)=\log_ax$ -- in fact the inverse of the exponential $a^x$ restricted to a suitable domain.
• Rational function. $f(x)=\dfrac{p(x)}{q(x)}$ for polynomials $p,q$ with $q(x)\ne0$; its domain is $R$ with the roots of $q$ removed.
• Reciprocal function. For $f(x)\ne0$, $g(x)=\dfrac1{f(x)}$ is the reciprocal of $f$, with domain $R$ minus the zeros of $f$; e.g. the largest domain of $f(x)=\dfrac1{x-1}$ is $R-\{1\}$.

Odd and even functions. $f:R\to R$ is odd if $f(-x)=-f(x)$ for all $x$; even if $f(-x)=f(x)$ for all $x$. $f(x)=x,\ 2x,\ x^3+2x$ are odd; $f(x)=x^2,\ 3,\ x^4+x^2,\ |x|$ are even; $f(x)=x+x^2$ is neither. …