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Recall that we introduced the set operation Cartesian product of two sets.
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Here we present the formal definition of a function. The definition from Wikipedia's Function page is as follows:
- Fun 1. For every $x \in A$ there exists $y \in B$ such that the pair $(x,y)$ belongs to $f$.
- Fun 2. For all pairs $(x_1,y_1)$ and $(x_2,y_2)$ belonging to $f$ the following implication holds: $x_1= x_2$ implies $y_1= y_2$.
- A function $f$ from $A$ to $B$ is often denoted by $f: A \to B$. If $f$ is a function then instead of $(x,y) \in f$ we write $y = f(x)$. The set $A$ is called the domain of $f$ and the set $B$ is called the codomain of $f$. The set \[ \bigl\{ y \in B \ : \ y = f(x) \ \text{for some} \ x \in A \bigr\} \] is called the range of $f$.
- If $A$ and $B$ are nonempty subsets of $\mathbb R$ we can rephrase the definition of a function as follows: A subset $f$ of the Cartesian product $\mathbb R\!\times\!\mathbb R$ is a function from $A$ to $B$ if every vertical line that goes through a point in $A\!\times\!\{0\}$ crosses $f$ at exactly one point. This formulation is sometimes called the vertical line test.
- In precalculus and calculus classes functions are often defined by formulas and the sets $A$ and $B$ are not specified explicitly. If this is the case, we consider that the domain of such a function is the largest subset of $\mathbb R$ on which the formula is defined. This domain is called the natural domain of a function. For example, the natural domain of the reciprocal function $r(x) = 1/x$ is the set ${\mathbb R}\!\setminus\!\{0\}$.
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Let us talk about a flip in this item.
- There is an important operation defined on a Cartesian product $A\!\times\!B$. That is the operation that flips the coordinates in $(x,y) \in A\!\times\!B$. Call this operation flip. Thus the flip of $(x,y) \in A\!\times\!B$ is the ordered pair $(y,x) \in B\!\times\!A$.
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Let $f:A\to B$ be a function. Consider the flip of $f$, call it $g$:
\[ g = \bigl\{ (y,x) \in B\!\times\!A \ : \ (x,y) \in A\!\times\!B \bigr\}. \]
A natural question is: Is $g$ a function from $B$ to $A$? Clearly $g$ is a subset of $B\!\times\!A$. So, for $g$ to be a function it has to satisfy two conditions in the definition of a function. Those conditions are:
- Flip 1. For every $y \in B$ there exists $x \in A$ such that the pair $(y,x)$ belongs to $g$.
- Flip 2. For all pairs $(y_1,x_1)$ and $(y_2,x_2)$ belonging to $g$ the following implication holds: $y_1 = y_2$ implies $x_1= x_2$.
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Notice that by the definition of $g$ we have that $(y,x) \in g$ if and only if $(x,y)\in f$. Therefore the conditions Flip 1 and Flip 2 can be reformulated in terms of the given function $f$ in the following way:
- Inv 1. For every $y \in B$ there exists $x \in A$ such that the pair $(x,y)$ belongs to $f$.
- Inv 2. For all pairs $(x_1,y_1)$ and $(x_2,y_2)$ belonging to $f$ the following implication holds: $y_1 = y_2$ implies $x_1 = x_2$.
- Notice that condition Inv 1 in traditional notation for functions can be expressed as: For every $y \in B$ there exists $x \in A$ such that $y = f(x)$. In other words, condition Inv 1 states that the range of $f$ equals $B$.
- Notice that condition Inv 2 in traditional notation for functions can be expressed as: If $x_1, x_2 \in A$ and $x_1 \neq x_2$, then $f(x_1) \neq f(x_2)$. In other words, condition Inv 2 states that the distinct elements in the domain of $f$ have distinct images in $B$.
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The reasoning in the previous item is the motivation for the following definition:
Definition. A function $f : A \to B$ from $A$ to $B$ is said to be invertible if it satisfies conditions Inv 1 and Inv 2. The function $g$ (that is the flip of $f$) \[ g = \bigl\{ (y,x) \in B\!\times\!A \ : \ (x,y) \in A\!\times\!B \bigr\} \] is called the inverse of $f$.
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Conditions Inv 1 and Inv 2 taken separately are motivation for the following two definitions.
Definition. A function $f$ from $A$ to $B$ is called surjection if it satisfies condition Inv 1.Definition. A function $f$ from $A$ to $B$ is called injection if it satisfies condition Inv 2. In other words, $f$ is injection if $x_1, x_2 \in A$ and $x_1 \neq x_2$ implies $f(x_1) \neq f(x_2)$.
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Together with the terminology injection and surjection goes the following terminology
Definition. A function $f$ from $A$ to $B$ is called bijection if it is both an injection and a surjection.
- Clearly, the terms bijection and invertible function are synonyms.
- A synonym for "injection" is "one-to-one function". A synonym for "surjection" is "onto function". I strongly encourage you to ignore these synonyms.
- If $A$ and $B$ are nonempty subsets of $\mathbb R$ we can rephrase the definition of an injection as follows: A function $f : A \to B$ is an injection if every horizontal line that goes through a point in $\{0\}\!\times\!B$ crosses $f$ at most one point.
- If $A$ and $B$ are nonempty subsets of $\mathbb R$ we can rephrase the definition of a surjection as follows: A function $f:A \to B$ is a surjection if every horizontal line that goes through a point in $\{0\}\!\times\!B$ crosses $f$ at least one point.
- If $A$ and $B$ are nonempty subsets of $\mathbb R$ we can rephrase the definition a bijection as follows: A function $f:A \to B$ is a bijection if every horizontal line that goes through a point in $\{0\}\!\times\!B$ crosses $f$ at exactly one point.
- The formulations in the preceding three bullets are sometimes called the horizontal line tests.
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In Precalculus you learned about some famous invertible functions. Let us start with affine functions.
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Below I plot several affine bijections and their inverses. As before, the functions are in navy blue and their inverses
are plotted in maroon. For each example you can guess what the values of $m$ and $b$ are used.
- Now to more complicated important invertible functions. Most functions in Precalculus are not invertible. For such functions we always identify their significant part (or restriction) which are invertible. Below I list several invertible functions and you can figure out why they are important: \begin{alignat*}{2} A & = [0,+\infty), & \quad B & = [0,+\infty), & \qquad f_1 & = \bigl\{ (x,x^2) \ : \ x \in A \bigr\} \\ A & = \mathbb{R}, & \quad B & = \mathbb{R}, & \qquad f_2 & = \bigl\{ (x,x^3) \ : \ x \in A \bigr\} \\ A & = \mathbb{R}, & \quad B & = (0,+\infty), & \qquad f_3 & = \bigl\{ (x,e^x) \ : \ x \in A \bigr\} \\ A & = (-\pi/2,\pi/2), & \quad B & = \mathbb R, & \qquad f_4 & = \bigl\{ \bigl( x,\tan(x) \bigr) \ : \ x \in A \bigr\} \\ A & = [-\pi/2,\pi/2], & \quad B & = [-1,1], & \qquad f_5 & = \bigl\{ \bigl(x,\sin(x)\bigr) \ : \ x \in A \bigr\} \\ A & = [0,\pi], & \quad B & = [-1,1], & \qquad f_6 & = \bigl\{ \bigl(x,\cos(x)\bigr) \ : \ x \in A \bigr\} \\ \end{alignat*}
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These six bijections are plotted below in navy blue. Their inverses are plotted in maroon.
$A = [0,+\infty), \ B = [0,+\infty), \ f_1 = \bigl\{ (x,x^2) \ : \ x \in A \bigr\}$
$A = \mathbb R, \ B = \mathbb R, \ f_2 = \bigl\{ (x,x^3) \ : \ x \in A \bigr\}$
$A = \mathbb R, \ B = (0,+\infty), \ f_3 = \bigl\{ (x,e^x) \ : \ x \in A \bigr\}$
$A = (-\pi/2,\pi/2), \ B = \mathbb R, \ f_4 = \bigl\{ \bigl( x,\tan(x) \bigr) \ : \ x \in A \bigr\}$
$A = [-\pi/2,\pi/2], \ B = [-1,1], \ f_5 = \bigl\{ \bigl(x,\sin(x)\bigr) \ : \ x \in A \bigr\}$
$A = [0,\pi], \ B = [-1,1], \ f_6 = \bigl\{ \bigl(x,\cos(x)\bigr) \ : \ x \in A \bigr\}$
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We also briefly discussed hyperbolic functions.
The most important hyperbolic functions are Sinh, Cosh and Tanh. They are defined as follows:
$\color{blue}{\cosh(x)} = \color{green}{\dfrac{1}{2} e^x} + \color{red}{\dfrac{1}{2} e^{-x}}, \ x \in \mathbb{R}$
$\color{blue}{\sinh(x)} = \color{green}{\dfrac{1}{2} e^x} \color{red}{- \dfrac{1}{2} e^{-x}}, \ x \in \mathbb{R}$
$ \tanh(x) = \dfrac{\sinh(x)}{\cosh(x)} = \dfrac{e^x - e^{-x}}{e^x + e^{-x}}, \ x \in \mathbb{R}$
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There are several reasons why hyperbolic functions are important. One reason is that they appear naturally in
important applications. Since we can not go into these applications in this class, I should offer a simpler reasons.
Here are some.
- From your mathematical experience in high school you are aware that the exponential function $f(x) = \exp(x)$ is important. One could argue that a drawback to the exponential function is that it is neither even, not odd function. So, one could ask: Does there exist an even function, call it $\operatorname{ev}(x)$, and an odd function, call it $\operatorname{od}(x)$, such that \[ \exp(x) = \operatorname{ev}(x) + \operatorname{od}(x) \quad \text{for all} \quad x \in \mathbb{R}. \] The answer to this question is yes and we can calculate these two functions. Since the equality that we want holds for all $x \in \mathbb R$ we can replace $x$ by $-x$ and still the equality will hold: \[ \exp(-x) = \operatorname{ev}(-x) + \operatorname{od}(-x) \quad \text{for all} \quad x \in \mathbb{R}. \] But, not remember that $\operatorname{ev}$ is an even function, so $\operatorname{ev}(-x) = \operatorname{ev}(x)$ and $o$ is an odd function, so $\operatorname{od}(-x) = -\operatorname{od}(x)$. Hence, the last equality can be rewritten as \[ \exp(-x) = \operatorname{ev}(x) - \operatorname{od}(x) \quad \text{for all} \quad x \in \mathbb R. \] Now adding the first equality and the last equality we get \[ \exp(x) + \exp(-x) = 2 \operatorname{ev}(x) \quad \text{for all} \quad x \in \mathbb R. \] Hence, $\operatorname{ev}(x) = \bigl(\exp(x) + \exp(-x)\bigr)/2 = \cosh(x)$. Similarly, subtracting the equality $\exp(-x) = \operatorname{ev}(-x) + \operatorname{od}(-x)$ from the equality $\exp(x) = \operatorname{ev}(x) + \operatorname{od}(x)$ we get \[ \exp(x) - \exp(-x) = 2 \operatorname{od}(x) \quad \text{for all} \quad x \in \mathbb{R}. \] Hence $\operatorname{od}(x) = \bigl(\exp(x) - \exp(-x)\bigr)/2 = \sinh(x)$. Thus, hyperbolic cosine and hyperbolic sine form a unique pair of an even and an odd function whose sum is the most important function: the exponential function.
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Why the names hyperbolic cosine and hyperbolic sine? Calculate the expression
\[
\bigl(\cosh(x)\bigr)^2 - \bigl(\sinh(x)\bigr)^2.
\]
You will see that
\[
\bigl(\cosh(x)\bigr)^2 - \bigl(\sinh(x)\bigr)^2 = 1 \quad \text{for all} \quad x\in \mathbb R.
\]
Now, I am not sure whether you learned in high school that the set
\[
\bigl\{ (x,y) \in \mathbb R\!\times \!\mathbb R \, : \, x^2 - y^2 = 1\bigr\}
\]
is called hyperbola. Here is its graph
Thus, by setting \[ \bigl\{ (\cosh(t), \sinh(t)) \in \mathbb R\!\times \!\mathbb R \, : \ t \in \mathbb R\bigr\} \] we get the right branch of the above hyperbola and by setting \[ \bigl\{ (-\cosh(t), \sinh(t)) \in \mathbb R\!\times \!\mathbb R \, : \ t \in \mathbb R\bigr\} \] we get the left branch of the above hyperbola. This is analogous to the fact that the set \[ \bigl\{ (\cos(t), \sin(t)) \in \mathbb R\!\times \!\mathbb R \, : \ t \in [0, 2 \pi) \bigr\} \] equals to the unit circle, since \[ \bigl(\cos(x)\bigr)^2 + \bigl(\sin(x)\bigr)^2 = 1. \]
Hyperbola $\bigl\{ (x,y) \in \mathbb R\!\times \!\mathbb R \, : \, x^2 - y^2 = 1\bigr\}$
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Consider a pair of functions $f : \mathbb R \to \mathbb R$ and $g : \mathbb R \to \mathbb R$ and ask the
following question:
Change the question slightly.
- It is important to point out that there are identities for hyperbolic functions which are quite similar to identities that you learned for trigonometric functions. For example, for all $x,y \in \mathbb R$ we have \begin{align*} \sinh(x + y) &= \sinh (x) \cosh (y) + \cosh (x) \sinh (y) \\ \cosh(x + y) &= \cosh (x) \cosh (y) + \sinh (x) \sinh (y) \\ \tanh(x + y) &= \frac{\tanh(x) +\tanh(y)}{1+ \tanh(x) \tanh(y) } \\ \end{align*} Many more properties of the hyperbolic functions you can find at the Wikipedia page Hyperbolic functions.
- I found an old book The Application of Hyperbolic Functions to Electrical Engineering Problems by Arthur Edwin Kennell. It is now a free ebook.
- We also defined four often neglected piecewise defined functions:
The sign function
The unit step function
The floor function
The ceiling function
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Here is a problem related to the concepts of a function and a bijection.
- Is the subset $F$ a function from ${\mathbb R}$ to ${\mathbb R}$? Give a detailed answer to this question. Does $F$ satisfy Fun 1? Does $F$ satisfy Fun 2?
- Find the largest subsets $A\subseteq {\mathbb R}$ and $B\subseteq {\mathbb R}$ such that the subset \[ f = \bigl\{(x,y) \in F \, : \, x\in A, y \in B \bigr\} \] of $A\!\times\!B$ is a surjection from $A$ to $B$. Find all such sets $A$ and $B$.
- Find the largest subsets $A\subseteq {\mathbb R}$ and $B\subseteq {\mathbb R}$ such that the subset \[ g = \bigl\{(x,y) \in F \, : \, x\in A, y \in B \bigr\} \] is a bijection from $A$ to $B$. Find all such sets $A$ and $B$.
The Unit Circle $\bigl\{ (x,y) \in {\mathbb R}\times {\mathbb R} \, : \, x^2+y^2=1 \bigr\}$
- Below is a formal definition of a composition of two functions.
Place the cursor over the image to start the animation.
The composition at a generic point $x$
The composition at another generic point $x$
Finding a minimum of the composition
Finding a maximum of the composition
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The animation below illustrates the maroon composition $g\circ
f$ of the navy blue function $f$ and the its inverse dark green function $g$.
Place the cursor over the image to start the animation.
