Piecewise Function

Piecewise functions are defined by different formulas on different parts of the domain.

Absolute Value Function

$|x|=\{\begin{array}{ll}x,& x\ge 0\\ -x,& x<0\end{array}$

Signum Function

$sgn(x)=\{\begin{array}{ll}x,& x\ge 0\\ -x,& x<0\end{array}$

Ramp Function

$R(x)=\{\begin{array}{ll}x,& x\ge 0\\ 0,& x<0\end{array}$

Floor and Ceiling Functions

Fractional-Part Function

Heaviside Function (Unit Step Function)

Basic Function $H(t)=\{\begin{array}{ll}0,& t<0\\ 1,& t\ge 0\end{array}$ Advanced Manipulations $H(t-a)=\{\begin{array}{ll}0,& t<a\\ 1,& t\ge a\end{array}$

\leq t \lt b\\ 0, & t \geq b \end{cases}$$ $$1 - H(t-a) = \begin{cases} 1, & t \lt a\\ 0, &t\geq a\end{cases}$$ Notice that $$f(t)H(t) = \begin{cases} 0, & t \lt 0\\ f(t), &t\geq 0\end{cases}$$ > [!example] Express piecewise as heaviside function > > Express the following function as a heaviside function. > > $$f(x) = \begin{cases} e^{-x} & x \lt0\\ 1-x^{2}, & 0\leq x \lt 1\\ \ln x & x \geq 1\end{cases}$$ > > Answer: > $$f(x) = e^{-x} (1-H(x)) + (1-x^{2})(H(x)- H(x-1))+ \ln x(H(x-1))$$