**Preface**

This series is aimed at providing tools for an electrical engineer to analyze data and solve problems in design. The focus is on applying calculus to equations or physical systems.

**Introduction**

This article will introduce functions.

There are many calculus references, the one I like to use is Calculus by Larson, Edwards and Hostetler.

This also assumes you are familiar with Python or can stumble your way through it.

**Concepts***Variable: *a symbol used to hold a value or coordinate.

*Set: * a combination of variables. In the form of *(a,b,c,...,n)* and usually (x,y) and (x,y,z).

*Coordinate: *a particular value taken on by a set used to indicate the position of a point, line, or plane.

*Function: * a real-valued function *f* maps *X* onto *Y* such that for every coordinate in *X* there is one and one only output on *Y*; where *X* and *Y* are sets of real numbers such that *X=(a1,b1,c1,...,z1)* and *Y=(a2,b2,c2,...,z2)*.

*Vector: * A set of variables, often direction and magnitude.

*Intercept: *a coordinate at which one coordinate value in the set is zero (e.g. f(a,b,0,d) or g(0,b,c)).

*Intersection: *a coordinate at which two sets of data (or functions) intercept.

*Graph: *a visual representation of coordinate sets, most often in 2 to 4 variables.

*Cartesian Coordinate System: a system in which the horizontal line on a graph is the x axis and the vertical is the y axis.*

*Slope: *rate of change of a function *m=(y _{2}-y_{1})/(x_{2}-x_{1}), x_{1} != x_{2}*

*Inverse: *A function g is the inverse of the function f if f(g(x)) = x for each x in the domain of g AND g(f(x)) = x for each x in the domain of f. All inverse functions are reflective about the line y=x.

**Vector Functions**

General Form | r(t)=f(t)i+g(t)j+h(t)k |

Addition | u(t) + v(t)=<u_{i} + v_{i}, u_{j} + v_{j}> |

Scalar Product | cu(t)=<cu_{i},cu_{j}> |

Dot Product | |

Cross Product | |

Commutative | u + v=v + u |

Associative | (u+v)+w=u+(v+w) |

Additive Identity | u+0=u |

Additive Inverse | u+(-u)=0 |

Distributive | (c+d)u=cu+du |

Distributive 2 | c(u+v)=cu+cv |

Scalar | 1(u)=u |

Scalar 2 | 0(u)=0 |

Where f, g and h are functions of t; r is a vector function of t; and i, j and k are vectors.

All the properties of the functions below can be components of a vector function. Therefore all the properties of the above properties apply to the functions below.

**Linear Functions**

General Form | Ax+By+C=0 |

Vertical Line | x=a |

Horizontal Line | y=b |

Point-slope Form | y-y_{1}=m(x-x_{1}) |

Slope-intercept Form | y=mx+b |

Where A, B, C, y_{1}, x_{1}a and b are coordinates; m is the slope; x and y are variables.

**Log Functions**

General Form | ln x = ∫^{b}_{a} (1/t) dt, for x>0 |

1 | ln(1) = 0 |

Multiple | ln(ab) = ln(a) + ln(b) |

Power | ln(a^{n}) = n ln a |

Division | ln(a/b) = ln(a) - ln(b) |

Where a, b and n are coordinates; x is a variable.

**Exponential Functions**

General Form | e^{x} = f^{-1}[ln(x)] |

Inverse 1 | ln(e^{x}) = x |

Inverse 2 | e^{ln(x)} = x |

Multiple | e^{a}e^{b}=e^{a+b} |

Division | e^{a}/e^{b}=e^{a-b} |

Where a, b and n are coordinates; x is a variable.

**Function Transformations**

Original Graph (Reference) | y=f(x) |

Horizontal Shift | y=f(x±c) |

Vertical Shift | y=f(x)±c |

Reflection (about x) | y=-f(x) |

Reflection (about y) | y=f(-x) |

Reflection (about origin) | y=-f(-x) |

**Axis Intercepts**

To determine where a function crosses the x or y axis evaluate the following function:

f(0) (for y-intercepts)

f(x)=0 (for x-intercepts)

If the above formula fails try using Newton's Method (not illustrated here).

**Next Up**

Continuity and Limits of functions.

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