MA101.4 Inverse Functions

Let f be a function and let $y=f(x), x \in D(f)$. We often call x the independent variable and y the dependent variable. Now consider specifying a number $y \in R(f)$ and working out what x’s are in the domain of the f, for which $f(x)=y$. There may of course be many such x’s corresponding to a particular y, but functions for which there is only ever one such x have a special name, they are called injections (their old name was one-to-one).

Definition

A function is an injection if for all x,y $f(x)=f(y) \Rightarrow x=y$

Or if you prefer $\forall x,y:\; x \neq y \Rightarrow f(x) \neq f(y)$

($\forall$ – for all and $\Rightarrow$ – implies.)

Injections are easily recognised from their graphs in that any line parallel to the x axis will meet the graph in at most one point.

Let f be an injection, then we use $f^{-1}(y), y \in R(y)$ to denote that single element $x \in D(f)$ which is such that $f(x)=y$.

The function $f^{-1} : R(f) \rightarrow D(f)$ is called the inverse function of the f.

By the definition of $f^{-1}$ we have, $D(f^{-1}) = R(f)$.

It is also clear that $R(f^{-1}) = D(f)$.

Note that functions that are not injections do not have inverse functions.

Example

Find the inverse function of the function f given by:

$f(x) = \begin{cases} (1 - x)^2 &\mbox{}0 \leq x \leq 1 \\ x+1 &\mbox{} 1 < x < 2 \end{cases}$

Solution: To have a sketch is a great help

$D(f)=[0,2)$ and $R(f)=[0,1] \cup (2,3)$

So $D(f^{-1}) = R(f) = [0,1] \cup (2,3)$.

Since the rule for f is given in two pieces we find the rule for $f^{-1}$ in two pieces.

$Let\; y \in [0,1] \\ So,\\ y=(1-x)^2 \\ y = 1 - 2x +x^2\\ x^2 - 2x + 1 - y = 0 \\ x = \frac{2-\sqrt{4-4(1-y)}}{2} \\ x = 1 - \sqrt{y}$

$Let\; y \in (2,3) \\So,\\ y = x+1 \; thus \; x = y-1$.

Whence $f^{-1}$ is given by

$f^{-1}(y) = \begin{cases} 1-\sqrt{y} &\mbox{}0 \leq y \leq 1 \\ y-1 &\mbox{} 2 < y < 3 \end{cases}$

Note that f is an injection and so $f^{-1}$ exists then the graphs f and $f^{-1}$ will be reflections of one another in the line y = x.

MA101.3 Sketching Functions

Graphs of functions may be sketched in the usual way.

Example

Sketch the graph of the function f given by:

$f(x) = \begin{cases} |x| &\mbox{}-2 < x \leq \pi \\ sin(x) -1 &\mbox{} \pi < x < \frac{3 \pi}{2} \\ -2x + 3\pi -2 &\mbox{} \frac{3 \pi}{2}

State the domain and range of f.

$D(f)=(-2,7)\backslash\{\frac{3\pi}{2}\}, R(f)=[0,\pi]\cup[-2,-1]\cup[3\pi-16,-2)$

Any set B which contains the range of a function (ie. $B \supseteq R(f)$) is called a co-domain of the f. The notation $f:A \rightarrow B$ (read: f from A to B) means simply f is a function whose domain is in  A (ie. $A=D(f)$) and whose range is in B (perhaps equal to B).

In practice mathematicians still refer to expressions in x as functions without mentioning any domain. When this is the case it is an unwritten law that one may assume the domain as the largest subset of the reals for which the expression is meaningful.

Example

What are the intended domains when we reference the functions

i) $\frac{1}{x^2-1}$

ii) $cot(x)$

iii) $\frac{x^2-1}{x-1}$

i) $\mathbb{R} \backslash \{-1,1\}$

ii) $\mathbb{R} \backslash \{ k \pi | k \in \mathbb{R} \}$

iii) $\mathbb{R} \backslash \{1\}$

MA101.2 Functions

The idea of a function is a rule that gives the mapping of an input number to an output number. This function is generally represented by $f(x)$ but nowadays we like the view that the f symbolises the function (intuitively f symbolises the rule the function is applying) and $f(x)$ is actually the number we get when we apply the function or the rule to the number $x$. (We say $f(x)$ is the value of f at x).

The modern idea of a function only allows rules which assign one value to a particular number of x. For example if f symbolised a rule for which say $f(2)=7$ and $f(2)=4$ then f is not a function. Do not confuse this with a rule f which is assigning the same value to two (or more different) numbers. For example $f(3)=5$ and $f(7)=5$; this is permitted in our idea of a function.

Also when specifying a function we must not only give the rule, but also give the set of numbers to which the rule is intended to apply. This set of numbers to which the rule applies is called the domain of the function and may be denoted by $D(f)$ or $D_f$. Thus we say, define the function f by: $f(x)=\frac{1}{x},x\in\mathbb{R}\backslash\{^-0\}$

Now let f be a function whose domain $D(f)=A$. Then the set of values that f takes ie $\{f(x)|x\in A\}$ is called the range of f and may be denoted by $R(f)$ or $R_f$

Some useful notation

• $(a,b) = \{ x \in \mathbb{R} | a < x < b \}$ – the open interval a,b
• $[a,b] = \{ x \in \mathbb{R} | a \leq x \leq b \}$ – the closed interval a,b
• $(a,b] = \{ x \in \mathbb{R} | a < x \leq b \}$
• $[a,b) = \{ x \in \mathbb{R} | a \leq x< b \}$
• $(\infty,a) = \{ x \in \mathbb{R} | x < a \}$
• $(\infty,a] = \{ x \in \mathbb{R} | x \leq a \}$
• $[-a,\infty) = \{ x \in \mathbb{R} | a \leq x \}$
• $(-a,\infty) = \{ x \in \mathbb{R} | a < x \}$

Also used is:

• $\mathbb{R}^+ = \{a \in \mathbb{R}, a > 0 \}$
• $\mathbb{R}^- =\{a \in \mathbb{R}, a < 0 \}$

$|x|$ (read this as mod x where $x \in \mathbb{R}$) is defined by:

$|x| = \begin{cases} x, & \mbox{if } x\geq \mbox{ 0} \\ -x, & \mbox{if } x<\mbox{0} \end{cases}$

In this course the notation $\sqrt{x}$ where $x \geq 0$ will always be the positive root, as will also $x^\frac{1}{2}$.

So, for example $\sqrt{4}=2$.

Note particularly the effect $\sqrt{(-2)^2}=\sqrt{4}=2$

In fact generally $\sqrt{x^2}=|x|$

The so called “floor function” (or integer part function) is notated as, $\lfloor x \rfloor$ and is defined by

$\lfloor x \rfloor = \begin{cases} \mbox{integer part of x}\\ \mbox{greatest integer less than or equal to x} \\ \mbox{the greatest integer not bigger than x} \end{cases}$

eg.

$\lfloor 5.7 \rfloor = 5$
$\lfloor 3 \rfloor = 3$
$\lfloor -2.7 \rfloor = -3$