Example 1 Divide \( x^3 + 3x^3 + 2x + 1 \) by \( x+2 \). Example 2 Perform a long division: \( (4x^4-6x^3+2x^2-3x+5) \div (2x+1) \).

# Functions

# Inverse Functions

Definition of Inverse Functions $$ \Large+ \leftarrow \text{ inverse operation } \rightarrow -$$ $$ \Large \times \leftarrow \text{ inverse operation } \rightarrow \div $$ $$ \Large x^2 \leftarrow \text{ inverse operation } \rightarrow \sqrt{x} $$ The function $y=4x-1$ can be undone by its inverse function $y=\dfrac{x+1}{4}$. We can consider of this act as two processes [...]

# Rational Functions

We have seen that a linear function has the form $y=mx+b$. When a linear function is devided by another function, the result is a rational function. Rational functions are characterised by asymptotes, which are lines the function gets close and close to but never reaches. The rational functions we consider can be written in the [...]

# Composite Functions

A composite function is formed from two functions in the following way. $$(g \circ f)(x) = g(f(x))$$ If $f(x)=x+3$ and $g(x)=2x$ are two functions, then we combine the two functions to form the composite function: \( \begin{align} (g \circ f)(x) &= g(f(x)) \\ &= 2f(x) \\ &= 2(x+3) \\ &= 2x+6 \end{align} \) That is, [...]

# Domain and Range

A relation may be described by: a listed set of ordered pairs a graph a rule The set of all first elements of a set of ordered pairs is known as the domain and the set of all second elements of a set of ordered pairs is known as the range. Alternatively, the domain is [...]

# Function Notation

Definition of Function Notation Consider the relation $y=3x+2$, which is a function. The $y$-values are determined from the $x$-values, so we say '$y$ is a function of $x$, which is abbreviated to $y=f(x)$. So, the rule $y=3x+2$ can be also be written as following. $$f: \mapsto 3x+2$$ $$\text{or}$$ $$f(x)=3x+2$$ $$\text{or}$$ $$y=3x+2$$ Function $f$ such that [...]

# Relations and Functions

Relations A relation is any set of points which connect two variables. A relation is often expressed in the form of an equation connecting the variables $x$ and $y$. In this case, the relation is a set of points $(x,y)$ in the Cartesian plane. This plane is separated into four quadrants according to the signs [...]