# Inverse Functions

Home > 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 [...]

# Rational Functions

Home > 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 [...]

# Composite Functions

Home > 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} [...]

# Domain and Range

Home > Functions 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, [...]

# Function Notation

Home > Functions 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 [...]