# Reciprocal Functions

These techniques involves sketching the graph of $y=\dfrac{1}{f(x)}$ from the graph of $y=f(x)$. Technique 1 When $f(x)$ approaches towards $0$, $y=\dfrac{1}{f(x)}$ approaches towards $\infty$, the graph of $y=\dfrac{1}{f(x)}$ approaches the vertical asymptote(s). Technique 2 The graph of $y=\dfrac{1}{f(x)}$ has vertical asymptotes at the $x$-intercepts of $y=f(x)$. Technique 3 When $f(x)$ approaches towards $\infty$, $y=\dfrac{1}{f(x)}$ approaches […]

# Inverse Functions – Ultimate Guide of Definition and Graphs

Domain and Range of Inverse Functions \Large \begin{align} + \leftarrow &\text{ inverse operation } \rightarrow – \\\times \leftarrow &\text{ inverse operation } \rightarrow \div \\x^2 \leftarrow &\text{ inverse operation } \rightarrow \sqrt{x} \end{align}The function $y=4x-1$ can be undone by its inverse function $y=\dfrac{x+1}{4}$.We can consider this act as two processes or machines. If […]

# 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 can be written in $y=\dfrac{ax+b}{cx+d}$. These functions have asymptotes which […]

# 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, $f(x)$ replaces $x$ in the function $g(x)$.The composite […]

# Domain and Range – An Ultimate Guide

A relation may be described by: 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 the set of independent values, and the range is the set […]

# 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 $x$, abbreviated to $y=f(x)$.So, the rule $y=3x+2$ can also be written as follows.\large \begin{align} f: \mapsto \ &3x+2 \\&\text{or} \\f(x)= \ &3x+2 \\&\text{or} \\y= \ &3x+2 \end{align}A function $f$ such that […]

# Relations and Functions

Relations A relation is any set of points that connect two variables.A relation is often expressed as an equation connecting the variables $x$ and $y$. The relation is a set of points $(x,y)$ in the Cartesian plane. This plane is separated into four quadrants according to the signs of $x$ and $y$. For example, $y=x+1$ […]