# Permissable to define a fork function with two x values mapping to different functions but whose y-values are identical?

Say we got this simple example

$$f(x) = \begin{cases} p_1(x),\text{ for }5 \leq x \leq 8,\\ p_2(x),\text{ for }8 \leq x \leq 10 \end{cases}$$

Assuming $$p_1(8)=p_2(8)$$, is this permissable?

Arguments for and against:

• Yes, because $$f(x)$$ maps each x-value to exactly one y-value.
• No, because it becomes unclear whether to choose $$p_1$$ or $$p_2$$ when $$x=8$$.
• Counter: well, it doesn't matter what function you choose, they yield identical y-values.
• The second bullet point doesn't make sense because $p1(8)=p_2(8)$ so the supposed "choice" is not actually there. – rubikscube09 Feb 25 at 19:48
• This is acceptable. – Ethan Bolker Feb 25 at 20:05

## 1 Answer

I would say it is an acceptable definition, in the sense its formalization describes a function. However, some people may be displeased, but again some people will always be displeased...

A function $$f:X\rightarrow Y$$ is a subset $$F$$ of $$X\times Y$$ such that:

• for every $$x\in X$$, there exists a $$y\in Y$$ for which $$(x,y)\in F$$;
• and, if $$(x, y_{1}), (x, y_{2})\in F$$, then $$y_{1}=y_{2}$$.

When we define a function by cases, that is not absolutely precise in the sense of this definition, but the correspondent idea can always be formalized: in your case, the set $$F$$ corresponding to the function may be defined as the subset of $$[5, 10]\times\mathbb{R}$$ (I am assuming here that your function is real-valued and not defined outside of $$[5,10]$$) such that $$F=\{(x,y)\in[5,10]\times\mathbb{R} : \text{if x\in[5,8], y=p_{1}(x), and if x\in [8,10], y=p_{2}(x)}\}.$$ Does that define a function?

• For every $$x\in [5,10]$$, there indeed exists an $$y\in\mathbb{R}$$ such that $$(x,y)\in F$$, equal to $$p_{1}(x)$$ if $$x\in[5,8)$$, equal to $$p_{2}(x)$$ if $$x\in(8,10]$$, and equal to $$p_{1}(x)=p_{2}(x)$$ if $$x=8$$.
• If $$(x, y_{1}), (x, y_{2})\in F$$, there are three cases: if $$x\in[5,8)$$, since $$p_{1}$$ is a function $$y_{1}=y_{2}$$; if $$x\in (8,10]$$, since $$p_{2}$$ is a function $$y_{1}=y_{2}$$; and if $$x=8$$, given both $$p_{1}$$ and $$p_{2}$$ are functions and $$p_{1}(8)=p_{2}(8)$$, $$y_{1}=y_{2}$$.

So $$F$$ indeed describes a function, and your definition is indeed acceptable.