Functions and Transformations of Functions
We will review some of the important concepts dealing with functions
and transformations of functions. Most likely you have encountered
each of these ideas previously, but here we will tie the concepts
together.
Definition of a Function
Let $A$ and $B$ be sets.
A function $F:A\to B$ is a relation that assigns to each
$x\in A$ a unique $y\in B$. We write $y=f(x)$ and call $y$ the
value of $f$ at $x$ or the image of $x$ under $f$.
We also say that $f$ maps $x$ to $y$.
The set $A$ is called the domain of $f$. The set of all
possible values of $f(x)$ in $B$ is called the range of $f$.
Here, we will only consider realvalued functions of a real variable,
so $A$ and $B$ will both be subsets of the real numbers. If $A$ is
left unspecified, we will assume it to be the largest set of real
numbers such that for all $x\in A$, $f(x)$ is real.
Examples
Click on a bullet to see the graph of that function as well as its
domain and range.
 • $f(x)=x^2$, $x$ real.
 • $f(x)=\sin x$, $x$ real.
 • $f(x)=\left\{
\begin{array}{rl}
1, & x < 0\\
2, & x\geq 0
\end{array}\right.$
 • $\displaystyle f(x)=\frac{1}{x+1}$.
 • $f(x)=\sqrt{x3}$.
Even/Odd Functions
A function $f:A\to B$ is said to be even if and only if
\[f(x)=f(x)\quad {\small\textrm{for all }} x\in A\]
and is said to be odd if and only if
\[f(x)=f(x)\quad {\small\textrm{for all }} x\in A.\]
Most functions are neither even nor odd.
The graph of an even
function is symmetric about the
yaxis, while the graph of an odd function is
symmetric about the origin.
Of the functions in the example,
 $f(x)=x^2$ is even.
 $f(x)=\sin x$ is odd.
 The others are neither even nor odd.
Transformations of Functions
We will examine four classes of transformations, each applied to the
function $f(x)=\sin x$ in the graphing examples.
 Horizontal translation: $g(x)=f(x+c)$.
The graph is translated $c$ units to the left if $c > 0$ and
$c$ units to the right if $c < 0$.
Graph
 Vertical translation: $g(x)=f(x)+k$.
The graph is translated $k$ units upward if $k > 0$ and
$k$ units downward if $k < 0$.
Graph
 Change of amplitude: $g(x)=Af(x)$.
The amplitude of the graph is increased by a factor of $A$ if $A > 1$
and decreased by a factor of $A$ if $A < 1$. In addition, if $A < 0$
the graph is inverted.
Graph
 Change of scale: $g(x)=f(ax)$.
The graph is "compressed" if $a > 1$ and "stretched out" if
$a < 1$. In addition, if $a < 0$ the graph is reflected about the
$y$axis.
Graph
In the Exploration, experiment with applying several transformations
to a single function.
Exploration
Key Concepts
A function F: $A\longrightarrow B$ is a relation that assigns to each $x
\in A$ a unique $y \in B$. We write $y = f(x)$ and call $y$ the value of $f$
at $x$ or the image of $x$ under $f$. We also say that $f$ maps $x$ to $y$.
The set $A$ is called the domain of $f$. The set of all possible values of
$f(x)$ in $B$ is called the range of $f$.
Each of these transformations takes a function $f$ and produces a new function $g$:
 Horizontal translation: $g(x) = f(x+c)$.
The graph is translated $c$ units to the left if $c > 0$ and $c$ units to
the right if $c < 0$.
 Vertical translation: $g(x) = f(x)+k$.
The graph is translated $k$ units upward if $k > 0$ and $k$ units
downward if $k < 0$.
 Change of amplitude: $g(x) = Af(x)$.
The amplitude of the graph is increased by a factor of $A$ if $A > 1$ and
decreased by a factor of $A$ if $A < 1$. In addition, if $A < 0$ the graph is
inverted.
 Change of scale: $g(x) = f(ax)$.
The graph is "compressed" if $a > 1$ and "stretched out" if $a < 1$.
In addition, if $a < 0$ the graph is reflected about the $y$axis.
