Common Functions and Asymptotic Behaviour

1. Power functions

$f(x)=x^p,x\ge 0,p\in \mathbb{R}$

• $p>0$, $f(x)$ increases as $x$ increases
• $p<0$, $f(x)$ decreases as $x$ increases
• $p=0$, $f(x)=1$ regardless of $x$
• $p=1$, $f(x)=x$

Key properties:

• $x^a\cdot x^b=x^{a+b}$
• $x^a/x^b=x^{a-b}$

2. Polynomial functions

$f(x)=a_0+a_{1}x+a_2{x}^2+\cdots +a_n{x}^n,\sum a_i{x}^i,i\in \mathbb{Z}$

• Behaviour for large $x$ determined by highest power
• $n$-th degree has $n$ roots

3. Exponential functions

$f(x)=b^x,b\in {\mathbb{R}}^{+},b\ne 1$

• Growth and decay
  • $b>1\Rightarrow f(x)$ exponential growth
  • $0<b<1\Rightarrow f(x)$ exponential decay
• Always positive
• Horizontal asymptote
• Base $e\Rightarrow$ the natural exponential function with $b=e$, also denoted $\exp (x)$, has special significance across mathematics

4. Logarithmic functions

$f(x)={\log }_b(x),\text{where }b\in {\mathbb{R}}^{+},b\ne 1,x\ge 0$

• Reverse exponentiation $\Rightarrow b^y=x⟺{\log }_b(x)=y$
• Growth and behaviour
  • $b>1$, ${\log }_b(x)$ increases as $x$ increases, but very slowly
  • $0<b<1$, ${\log }_b(x)$ decreases as $x$ increases
• Vertical asymptote at $x=0$
• ${\log }_b(1)=0$ and ${\log }_b(b)=1$
• $\ln (x)$ has $e$ as base

Product rule $\Rightarrow {\log }_b(xy)={\log }_b(x)+{\log }_b(y)$

Break up exponent $\Rightarrow {\log }_b(y^x)=x{\log }_b(y)$

${\log }_b(x)=\frac{{\log }_a(x)}{{\log }_a(b)}$