# Ramp Function

A Ramp Function is an Unary Function defined as [math]R(x) := \max(x,0) [/math]

**AKA:**Max Function.**Example(s):**- [math]R(x) := \begin{cases} x, & x \ge 0; \\ 0, & x\lt 0 \end{cases} [/math]
- [math]R(x) := \frac{x+|x|}{2} [/math]

**Counter-Example(s)****See:**Unary Function, Piecewise Function, Real Function, Graph of a Function, Scaling And Shifting.

## References

### 2018

- (Wikipedia, 2018) ⇒ https://en.wikipedia.org/wiki/Ramp_function Retrieved:2018-2-18.
- The
**ramp function**is a unary real function, whose graph is shaped like a ramp. It can be expressed by numerous definitions, for example "0 for negative inputs, output equals input for non-negative inputs". The term "ramp" can also be used for other functions obtained by scaling and shifting, and the function in this article is the*unit*ramp function (slope 1, starting at 0).This function has numerous applications in mathematics and engineering, and goes by various names, depending on the context.

**Definitions**The ramp function (

*R*(*x*) : ℝ → ℝ_{0}^{+}) may be defined analytically in several ways. Possible definitions are:- A piecewise function:
- [math]R(x) := \begin{cases} x, & x \ge 0; \\ 0, & x\lt 0 \end{cases} [/math]

- The max function:
- [math]R(x) := \max(x,0) [/math]

- The mean of an independent variable and its absolute value (a straight line with unity gradient and its modulus):
- [math]R(x) := \frac{x+|x|}{2} [/math]

- A piecewise function:

- The

- this can be derived by noting the following definition of max(
*a*,*b*),- [math] \max(a,b) = \frac{a+b+|a-b|}{2} [/math]
- for which
*a*=*x*and*b*= 0

- The Heaviside step function multiplied by a straight line with unity gradient:

- [math]R\left( x \right) := xH(x)[/math]
- The convolution of the Heaviside step function with itself:
- [math]R\left( x \right) := H(x) * H(x)[/math]

- The integral of the Heaviside step function:
^{[1]}

- [math]R(x) := \int_{-\infty}^{x} H(\xi)\,d\xi[/math]

- [math]R(x) := \langle x\rangle[/math]

- this can be derived by noting the following definition of max(