Functions

• Inverse of Function
• Composition of Functions

Increasing/decreasing

• If a function increases/decreases the whole way, it's a monotonic function (1 to 1)
  • Every y-value is unique/has a unique x value
  • A horizontal/vertical line will intersect the function at most once
  • If $f$ is one-to-one, then $f$ has an Inverse of Function

Transformations

• Transformation Notation
• Invariant points <> Points that stay the same after a transformation

Types

• Exponential Functions

• Quadratics

• Discreet <> Functions that are just a bunch of points

• Piecewise linear <> Functions that are just a bunch of points connected with lines

Parity of Function

• Even functions have symmetry along the y axis

bottom=-1; top=14;
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y=x^{2}
y=(-x)^{2}|dashed|red

• Algebraically:
  • $f(x)=f(-x)$
• Odd functions have symmetry about the origin

y=\frac{1}{x}
y=\frac{-1}{x}|dotted|green
y=\frac{1}{-x}|dashed|red

• Algebraically:
  • $f(-x)=-f(x)$

• For any function $f(x)$, it can be written as the sum of an even and odd function
  • $$$$

f(x) & =E(x)+O(x)\tag{1} \label{a} \
f(-x) & =E(-x) +O(-x) \
& =E(x)-O(x)\tag{2} \label{b} \because \text{E is even, O is odd} \
f(x)+f(-x) & =2E(x) & \eqref{a}+\eqref{b} \
E(x) & =\frac{1}{2}(f(x)+f(-x)) \
O(x) & =\frac{1}{2}(f(x)-f(-x)) \
E(x)+O(x) & =f(x)
\end