Real functions

## Real functions
            ---
            ## Functions
            A function is a machine that takes in things and puts out things.
            Usually, the things are real numbers.

$f:X\to Y$ has

*domain* $X$, the set of inputs,

*codomain* $Y$, the set that all the outputs belong to,

*range* (or *image*) $f(X)=\\{y\in Y$ such that $y=f(x)$ for some $x\in X\\}$.

Can use a table, a rule, a formula to define a function.

| $x$  | \| | $2$  | $4$   | $6$   | $8$ | $10$ | $12$ |
            |-----|-|----|-----|-----|---|----|----|
            | $f(x)$ | \| | $13$ | $\pi$ | $\frac12$ | $1$ | $1$  | $3$  |
            
            ---
            ## Functions
            A function is a machine that takes in things and puts out things.
            Usually, the things are real numbers.

$f:X\to Y$ has

*domain* $X$, the set of inputs,

*codomain* $Y$, the set that all the outputs belong to,

*range* (or *image*) $f(X)=\\{y\in Y$ such that $y=f(x)$ for some $x\in X\\}$.

Can use a table, a rule, a formula to define a function.

$f(x) = x$ if $x$ is an integer, and $f(x) = t$ such that $t$ is the number of pixels used to render the number $x$ in this font, if $x$ is not an integer.
            
            ---
            ## Functions
            A function is a machine that takes in things and puts out things.
            Usually, the things are real numbers.

$f:X\to Y$ has

*domain* $X$, the set of inputs,

*codomain* $Y$, the set that all the outputs belong to,

*range* (or *image*) $f(X)=\\{y\in Y$ such that $y=f(x)$ for some $x\in X\\}$.

Can use a table, a rule, a formula to define a function.

$f(x) = x^2 - \sin(x)$
            
            ---
            ## Functions
            A function is a machine that takes in things and puts out things.
            Usually, the things are real numbers.

$f:X\to Y$ has

*domain* $X$, the set of inputs,

*codomain* $Y$, the set that all the outputs belong to,

*range* (or *image*) $f(X)=\\{y\in Y$ such that $y=f(x)$ for some $x\in X\\}$.

Can use a table, a rule, a formula to define a function.

$f(x) = x^2 - \sin(x)$

Most of our functions will be defined by formulae.
            
            ---
            ## Functions
            A function is a machine that takes in things and puts out things.
            Usually, the things are real numbers.

$f$ is the function, $f(x)$ means "the function $f$ evaluated at the input $x$" means "the output of $f$ which corresponds to the input $x$".

A function always gives the same output for the same input.
            If $x$ is fixed, then $f(x)$ is fixed.
            
            ---
            ## Functions
            A function is a machine that takes in things and puts out things.
            Usually, the things are real numbers.

A function always gives the same output for the same input.
            If $x$ is fixed, then $f(x)$ is fixed.

But different inputs could give the same output.

#### Example
            $f:\mathbb R\to\mathbb R$ defined by $f(x) = x^2$ has $f(-x)=f(x)$ for all $x$, so $f$ is not injective.

#### Definition
            A function $f$ is *injective* if $f(x_1)=f(x_2) \implies x_1=x_2$.
            
            ---
            ## Functions: injectivity and surjectivity
            A function is a machine that takes in things and puts out things.
            Usually, the things are real numbers.

A function always gives the same output for the same input.
            If $x$ is fixed, then $f(x)$ is fixed.

#### Definition
            A function $f$ is *injective* if $f(x_1)=f(x_2) \implies x_1=x_2$.

A function $f:X\to Y$ is *surjective* if its range is its codomain; if $\\,\forall y \in Y,\\;\exists\\,x \in X$ such that $f(x)=y$.

A function is *bijective* if it is both injective and surjective.
            
            ---
            ## Functions: injectivity and surjectivity

#### Definition
            A function $f$ is *injective* if $f(x_1)=f(x_2) \implies x_1=x_2$.

A function $f:X\to Y$ is *surjective* if its range is its codomain; if $\\,\forall y \in Y,\\;\exists\\,x \in X$ such that $f(x)=y$.

A function is *bijective* if it is both injective and surjective.
            #### Examples
            Which of the following are injective, surjective, bijective on domain $\mathbb R$ and codomain $\mathbb R$?

$f(x) = x^3, \\;\\; g(x) = x^3-x, \\;\\; h(x) = \frac1{\lvert x \rvert +1}, \\;\\; p(x) = \sin(x)$,

$q(x) = 5, \\;\\; r(x) = x+\lceil x \rceil$.

(How) can you restrict their domains and codomains to make them bijective?
            
            ---
            ## Functions: injectivity and surjectivity

#### Definition
            A function $f$ is *injective* if $f(x_1)=f(x_2) \implies x_1=x_2$.

A function $f:X\to Y$ is *surjective* if its range is its codomain; if $\\,\forall y \in Y,\\;\exists\\,x \in X$ such that $f(x)=y$.

A function is *bijective* if it is both injective and surjective.

#### Examples
            $f(x) = x^3$.

Every horizontal line crosses the curve $\leq1$ time, so $f$ is injective.

Every horizontal line crosses the curve $\geq1$ time, so $f$ is surjective.

$f$ is both injective and surjective, so bijective.
            </div>
            <div style="flex:1;margin-left:2vw">

![Desmos plot of y=x cubed](gfx/x3.png "Desmos plot of y=x^3")
            </div>
            </div>

---
            ## Functions: injectivity and surjectivity

#### Definition
            A function $f$ is *injective* if $f(x_1)=f(x_2) \implies x_1=x_2$.

A function $f:X\to Y$ is *surjective* if its range is its codomain; if $\\,\forall y \in Y,\\;\exists\\,x \in X$ such that $f(x)=y$.

A function is *bijective* if it is both injective and surjective.

#### Examples
            $g(x) = x^3-x$.

Some horizontal lines cross the curve $>1$ time, so $g$ is not injective.

Every horizontal line crosses the curve $\geq1$ time, so $g$ is surjective.

$g$ is not injective, so it is not bijective.
            </div>
            <div style="flex:1;margin-left:2vw">

![Desmos plot of y=x cubed minus x](gfx/x3-x.png "Desmos plot of y=x^3-x")
            </div>
            </div>

---
            ## Functions: injectivity and surjectivity

#### Definition
            A function $f$ is *injective* if $f(x_1)=f(x_2) \implies x_1=x_2$.

A function $f:X\to Y$ is *surjective* if its range is its codomain; if $\\,\forall y \in Y,\\;\exists\\,x \in X$ such that $f(x)=y$.

A function is *bijective* if it is both injective and surjective.

#### Examples
            $h:\mathbb R\to\mathbb R$ given by $h(x) = \frac1{\lvert x \rvert +1}$
            is neither injective nor surjective.
            But

$h:[0,\infty)\to(0,1]$ would be bijective.

$h:(-\infty,0]\to(0,1]$ would be bijective.
            </div>
            <div style="flex:1;margin-left:2vw">

![Desmos plot of y=1 on (abs x plus 1)](gfx/1onabsxplus1.png "Desmos plot of y=1/(|x|+1)")
            </div>
            </div>

---
            ## Functions: injectivity and surjectivity

#### Definition
            A function $f$ is *injective* if $f(x_1)=f(x_2) \implies x_1=x_2$.

A function $f:X\to Y$ is *surjective* if its range is its codomain; if $\\,\forall y \in Y,\\;\exists\\,x \in X$ such that $f(x)=y$.

A function is *bijective* if it is both injective and surjective.

#### Examples
            $p:\mathbb R\to\mathbb R$ given by $p(x) = \sin(x)$
            is neither injective nor surjective.
            But

$p:[\frac{-\pi}2,\frac\pi2]\to[-1,1]$ would be bijective.

$p:[\frac{-3\pi}2,\frac{-\pi}2]\to[-1,1]$ would be bijective.

$p:[-\pi,0]\to[-1,0]$ would be surjective but not injective.
            </div>
            <div style="flex:1;margin-left:2vw">

![Desmos plot of y=sin(x)](gfx/sinx.png "Desmos plot of y=sin(x)")
            </div>
            </div>

---
            ## Functions: injectivity and surjectivity

#### Definition
            A function $f$ is *injective* if $f(x_1)=f(x_2) \implies x_1=x_2$.

A function $f:X\to Y$ is *surjective* if its range is its codomain; if $\\,\forall y \in Y,\\;\exists\\,x \in X$ such that $f(x)=y$.

A function is *bijective* if it is both injective and surjective.

#### Examples
            $q:\mathbb R\to\mathbb R$ given by $q(x) = 5$
            is neither injective nor surjective.

To make $q$ bijective, we would have to its domain and codomain to single points.
            </div>
            <div style="flex:1;margin-left:2vw">

</div>
            </div>

---
            ## Functions: injectivity and surjectivity

#### Definition
            A function $f$ is *injective* if $f(x_1)=f(x_2) \implies x_1=x_2$.

A function $f:X\to Y$ is *surjective* if its range is its codomain; if $\\,\forall y \in Y,\\;\exists\\,x \in X$ such that $f(x)=y$.

A function is *bijective* if it is both injective and surjective.

#### Examples
            $r:\mathbb R\to\mathbb R$ given by $r(x) = x+\lceil x\rceil$
            is injective but not surjective.
            But

$r:(0,1]\to(1,2]$ would be bijective.

$r:(0,2]\to(1,2]\cup(3,4]$ is bijective.

$r:(0,2]\to(1,4]$ is injective but not surjective.
            </div>
            <div style="flex:1;margin-left:2vw">

![Desmos plot of y=x plus the ceiling function of x](gfx/xplusceilx.png "Desmos plot of y=x+ceil(x)")
            </div>
            </div>