Functions

Overview

A mapping takes values from a given input set and maps each of them to one or more output values. For example, is a mapping:

A function is a mapping where every value maps to a single output value. Functions are defined by a rule from input to output, and a set for the domain. Functions can be one-to-one, or many-to-one. Functions can be shown by the vertical line test, while one-to-one functions are shown by the horizontal line test.

Definitions:

The image of an input value is the corresponding output value of the function.
The domain is the set of possible input values to the function.
The range is the set of possible output values of the function.
The composite function formed by applying to and then applying to the result is denoted , , or .
The notation represents applied to times, e.g. , apart from trig functions.

Inverse functions

The inverse function of is , such that:

The inverse function only exists for one-to-one functions. To find the inverse function, start with , rearrange for , and replace the s with .

The domain of is the same as the range of . The range of is the same as the domain of . Graphically, the graph of is a reflection of in the line .

Graph transformations

Combining graph transformations of the and axis can be done by considering each axis individually. However, combing graph transformations on the same axis is different for both axes.

-axis translations and stretches

For the -axis, transforms the graph of by first stretching it by scale factor parallel to the -axis, then translating by units in the positive direction.

-axis combined transformations

For the -axis, transforms the graph of by first translating by units in the negative direction, then stretching by scale factor parallel to the axis. This can be thought of as first replacing the with (translating), then replacing the in with (stretching.)

Modulus functions

The modulus function reflects parts of a graph below the axis to be above the axis. It is defined as:

Graph

The graph of has a 'V' shape, with a gradient of , -intercept of , and a vertex at .

Partial fractions

A rational function cam be expressed as an algebraic fraction where both the numerator and denominator are polynomials e.g..

A proper rational function is where the degree of the numerator is strictly less than the degree of the denominator. To make an improper rational function proper, polynomial division can be used:

A proper rational function can be decomposed into partial fractions:

Note that repeated factors require a fraction with a denominator of the single factor, and a fraction with a denominator of the repeated factor.