What does this $f(x) = \{$ mean?

Question

Let $f:\mathbb{Z}\to\mathbb{Z}$ be defined by

$$f(x)=\begin{cases}\frac{x}{2}&x \text{ is even}\\ 0&x \text{ is odd}\end{cases}$$

Does this mean for all values of $f(x) = x/2, x$ is even and for all values of $f(x) = 0,x$ is odd?

Answer 1

Yes: if, say, $P(x)$, $Q(x)$ and $R(x)$ are predicates in the variable $x$, then $$f(x)=\begin{cases}g(x)&\text{if }P(x)\\ h(x)&\text{if }Q(x)\\ i(x)&\text{if }R(x)\end{cases}$$

is standard notaton for a function $f$ such that $f(x)=g(x)$ for all $x$ such that $P(x)$, $f(x)=h(x)$ for all $x$ such that $Q(x)$, and $f(x)=i(x)$ for all $x$ such that $Q(x)$. And so on for more cases. This kind of presentation of a function is called piecewise definition, and the "if" is often omitted because that is the way the cookie crumbles. It is highly recommendend, if not mandatory, that the cases listed cover the entirety of the domain of $f$. It is good practice to also have the cases listed be mutually exclusive. Often, the last case listed may be substituted by the word "otherwise" instead of actually writing down the negation of the logical disjunction of all the other cases.

Answer 2

It means that if $x$ is even, hence $x=2k$ for some $k\in\mathbb{Z}$, then $f(x)=k$. Otherwise, if $x$ is odd then $f(x)=0$

Answer 3

For any even integer number $x$ i.e $....,-2, 2 , 4 ,6 ..... $ the function divides $x$ by 2.

For any odd integer number $x$ i.e $....,-1, 1 , 3 ,5 ..... $ the function output is zero.

Answer 4

First, note that this function has as its domain and contradiction the set of integers.

The expression $$f(x)=\begin{cases}\dfrac{x}{2}&x \text{ is even}\\[3pt] 0&x \text{ is odd}\end{cases}$$

It means that if $x$ is even, then the value of $f(x)$ will be $x/2$. For example, $f(10)=10/2=5$, $f(2)=2/2=1$, if $x$ is odd, then $f(x)$ is $0$, for example $f(1)=0, f(3)=0, f(21)=0$, ...

This is a piecewise function

Answer 5

Admittedly, this would be more legible perhaps when written $$ f(x)=\begin{cases}\frac x2&\text{if $x$ is even}\\0&\text{if $x$ is odd}\end{cases}$$ but the short is, well, shorter.

This is the unique function with the properties

• If $x$ is even, then $f(x)=\frac x2$
• If $x$ is odd, then $f(x)=0$.

Note that $f(0)=0$ even though $0$ is even (cf. Numberphile).

It is important to note that for each $x$ in the domain, exactly one of the cases applies and hence a unique function value is defined, as is needed.

---

(Sometimes, you may encounter a piecewise definition with overlapping conditions, perhaps something like $$ |x|=\begin{cases}x&\text{if }x\ge 0\\-x&\text{if }x\le 0\end{cases}$$ in that case, one has to verify that different branches do give the same function value where they overlap)