+---
+geometry: margin=2cm
+author: Andrew Lorimer
+---
+
# Transformation
**Order of operations:** DRT - Dilations, Reflections, Translations
- $k$ - translation of $k$ units parallel to $y$-axis or from $x$-axis
- $h$ - translation of $h$ units parallel to $x$-axis or from $y$-axis
- for $(ax)^n$, dilation factor is $1 \over a$ parallel to $x$-axis or from $y$-axis
+- when $0 < |a| < 1$, graph becomes closer to axis
## Translations
| $n$ is even: | $n$ is odd: |
| ------------ | ----------- |
-|{#id .class width=50%} | {#id .class width=50%} |
+|{#id .class width=20%} | {#id .class width=20%} |
### $x^n$ where $n \in \mathbb{Z}^-$
| $n$ is even: | $n$ is odd: |
| ------------ | ----------- |
-|{#id .class width=50%} | {#id .class width=50%} |
+|{#id .class width=20%} | {#id .class width=20%} |
### $x^{1 \over n}$ where $n \in \mathbb{Z}^+$
| $n$ is even: | $n$ is odd: |
| ------------ | ----------- |
-|{#id .class width=50%} | {#id .class width=50%} |
+|{#id .class width=20%} | {#id .class width=20%} |
### $x^{-1 \over n}$ where $n \in \mathbb{Z}^+$
$$x^{p \over q} = \sqrt[q]{x^p}$$
-- if $p \gt q$, the shape of $x^p$ is dominant
-- if $p \lt q$, the shape of $x^{1 \over q}$ is dominant
+- if $p > q$, the shape of $x^p$ is dominant
+- if $p < q$, the shape of $x^{1 \over q}$ is dominant
- points $(0, 0)$ and $(1, 1)$ will always lie on graph
- Domain is: $\begin{cases} \mathbb{R} \hspace{4em}\text{ if }q\text{ is odd} \\ \mathbb{R}^+ \cup \{0\} \hspace{1em}\text{if }q\text{ is even}\end{cases}$
Addition of linear piecewise graphs - add $y$-values at key points
-Product functions:
+Product functions:
+
- product will equal 0 if one of the functions is equal to 0
- turning point on one function does not equate to turning point on product
Find new point $(x^\prime, y^\prime)$. Substitute these into original equation to find image with original variables $(x, y)$.
+## Composite functions
+
+$(f \circ g)(x)$ is defined iff $\operatorname{ran}(g) \subseteq \operatorname{dom}(f)$
+