# Intro to functions

Functions are a way of describing a special relationship between an input and an output variable.

It is like a set of rules for how to transform a number.

It is mainly written as f(x), such that x is the input vairable.

## Plotting a function

if the input variable represents the x-axis and the output the y axis of a coordinate plane then a function can be plotted on a graph. To do this by hand plot a 4-5 points on the graph and then join them with a smooth line.

## Vertical line test

A function takes a single input and give a single output. If there is more than one outcome the equation is not a function.

To test if an equation is a function, you can perform the verticla line test. To do this graph the function and move a vertical line, such as a ruler horizontally across the graph. If the vertical line touches more than one point, then these points share the same input and it is not a function. If no two points touch the vertical line at the same time the equation has passed the test and is a function

# Terms to describe functions

Term |
Examples |
Definition |

Monomial | 2x, m, 3, | A number, a variable or a product of numbers and variables. |

Polynomial | 2x+5, k^{2}+8 |
Any function that connects coefficents and variables only with the operations of multiplicaiton, devison, addition and subtraction. |

Degree of a polynomial | x^{4}+3x^{2}-2x |
The greatest exponent in a function with a variable base. |

Leading Coefficient | 3x, $\frac{\mathbf{1}}{\mathbf{4}}x$ |
Coefficient of the term with the highest exponent. |

Turning Point | A point on a graph where the gradient changes from positive to negative or from negative to positive. |

# Families of Functions

Name of function | General form | Example | Characteristics |

Linear | ax+b | 2x -1 | The highest power/exponent of any variable in the function is 1. |

Quadratic | ax^{2}+bx+c |
x^{2} + x -5 |
The highest power/exponent of any variable in the function is 2. |

Cubic | ax^{3}+bx^{2}+cx+d |
4x^3 -x^2 | The highest power/exponent of any variable in the function is 3. |

Rational | $\frac{P\left(x\right)}{Q\left(x\right)}$ | $\frac{2{x}^{2}+1}{x-3}$ | Devision is used as an operator |

Exponential | ${a}^{x-b}+c$ | f(x)=2^(x-1) -5 | There is a variable in the exponent. |

Logarithmic | ${\mathrm{log}}_{a}\left(P\left(x\right)\right)$ | f(x)=log(2x-3) | It has a log in it. |

Trigonometric | $a*trig(x-b)+c$ | f(x)=Cos(x) | It has a trigonometric ratio in it. |

# Linear functions

## Function

$f\left(x\right)=mx+c$

m = determines the gradient/steepness of the line

c = changes y-intercept

x = input number

## Functions for parallel and perpendicular lines

**Parallel Lines:** have same gradient but different y-intercepts

**Perpendicular line:** has a negative and flipped gradient (this also affect and determines the y-intercepts)

# Exponential Functions

## Function

$f\mathit{\left(}x\mathit{\right)}\mathit{=}a{b}^{zx\mathit{}\mathit{+}c}\mathit{+}d$

a= Determines reflection on x-axis

b= Changes the gradient/steepness of the line

c= Transforms the line on the x-axis

d= Changes horizontal asymptote and y-intercept

x= Input number

z= Determines reflection on y-axis.

# Domain and Range

**Definition:** The domain is the restrictions of the x values.

E.g. Domain is: x > 0, x € ℝ

## Range

The set of all possible outputs of a function (z-values on the graph) is called the range of a function.

E.g: Range is: 1 < y < 3, y = ℝ

Things that restrict the Domain and range:

- You cannot divide by zero.
- You cannot take the square root of a negative number.
- You cannot take the logarithm of a negative number or zero.

# Transforming functions

# Sin, Cos functions

Basic structure of the functions:

f(x)=a(x-h)2+k

f(x)=asin (bx)+c

# Editors

- Cerita - 390 words.
- joeClinton - 273 words.

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