73 lines
2.6 KiB
Markdown
73 lines
2.6 KiB
Markdown
# MATH 117: Calculus 1
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## Functions
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A **function** is a rule where each input has exactly one output, which can be determined by the **vertical line test**.
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!!! definition
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- The **domain** is the set of allowable independent values.
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- The **range** is the set of allowable dependent values.
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Functions can be **composed** to apply the result of one function to another.
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$$
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(f\circ g)(x) = f(g(x))
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$$
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!!! warning
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Composition is not commutative: $f\circ g \neq g\circ f$.
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## Inverse functions
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The inverse of a function swaps the domain and range of the original function: $f^{-1}(x)$ is the inverse of $f(x)$.. It can be determined by solving for the other variable:
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$$
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\begin{align*}
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y&=mx+b \\
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y-b&=mx \\
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x&=\frac{y-b}{m}
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\end{align*}
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$$
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Because the domain and range are simply swapped, the inverse function is just the original function reflected across the line $y=x$.
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<img src="https://upload.wikimedia.org/wikipedia/commons/1/11/Inverse_Function_Graph.png" width=300>(Source: Wikimedia Commons, public domain)</img>
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If the inverse of a function is applied to the original function, the original value is returned.
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$$f^{-1}(f(x)) = x$$
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A function is **invertible** only if it is "**one-to-one**": each output must have exactly one input. This can be tested via a horizontal line test of the original function.
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If a function is not invertible, restricting the domain may allow a **partial inverse** to be defined.
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!!! example
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<img src="https://upload.wikimedia.org/wikipedia/commons/7/70/Inverse_square_graph.svg">(Source: Wikimedia Commons, public domain)</img>
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By restricting the domain to $[0,\inf]$, the **multivalued inverse function** $y=\pm\sqrt{x}$ is reduced to just the partial inverse $y=\sqrt{x}$.
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## Symmetry
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An **even function** satisfies the property that $f(x)=f(-x)$, indicating that it is unchanged by a reflection across the y-axis.
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An **odd function** satisfies the property that $-f(x)=f(-x)$, indicating that it is unchanged by a 180° rotation about the origin.
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The following properties are always true for even and odd functions:
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- even × even = even
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- odd × odd = even
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- even × odd = odd
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Functions that are symmetric (that is, both $f(x)$ and $f(-x)$ exist) can be split into an even and odd component. Where $g(x)$ is the even component and $h(x)$ is the odd component:
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$$
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\begin{align*}
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f(x) &= g(x) + h(x) \\
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g(x) &= \frac{1}{2}(f(x) + f(-x)) \\
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h(x) &= \frac{1}{2}(f(x) - f(-x))
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\end{align*}
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$$
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!!! note
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The hyperbolic sine and cosine are the even and odd components of $f(x)=e^x$.
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$$
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\cosh x = \frac{1}{2}(e^x + e^{-x}) \\
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\sinh x = \frac{1}{2}(e^x - e^{-x})
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$$
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