Restrictions and extensions

Let \(A\) and \(B\) be sets and let \(f\) be a function from \(A\) to \(B\).

Let \(D \subseteq A\). The restriction of \(f\) to D, denoted by \(f|_D\), is the function from \(D\) to \(B\) such that \[\operatorname{gr}(f|_D) = \{ (x,y) \mid x \in D \wedge (x,y) \in \operatorname{gr}(f) \}.\]

For example, let \(f:\mathbb{R} \rightarrow \{0,1\}\) be given by \[f(x) = \left \{ \begin{array}{ll} 1 & \text{if } 0 \leq x \leq 1\\ 0 & \text{otherwise}.\end{array}\right.\]

Let \(D\) denote the set \(\{ x \in \mathbb{R} \mid 0 < x < 1 \}\). Then, \(f|_D\) is a constant function with \(f(x) = 1\) for every \(x \in D\).

Function restrictions are often found in the study of periodic functions or the local behavoriour of functions. These applications are more common in analysis courses than discrete math courses. We will revisit function restrictions when we look at inverse functions later.

Given two functions \(f\) and \(g\), \(g\) is said to be an extension of \(f\) if \(g\) is a restriction of \(f\). Function extensions are seen quite a bit in complex analysis but not so much in this course. You can learn more about complex analysis in MATH 3007 or MATH 3057.

Exercises

1. Let \(f:\mathbb{Z} \rightarrow \mathbb{Z}\) be given by \(f(x) = 3x^2 - 4\). Let \(D = \{-5,2,7\}\). Give the graph of \(f|_D\).

2. Let \(f:\mathbb{R} \rightarrow \mathbb{R}\) be given by \[f(x) = \left\{\begin{array}{ll} x^2 & \text{if } x < -1 \\ \frac{1}{x-2} & \text{if } -1 \leq x < 2 \\ 3-\sqrt{x} & \text{if } 2 \leq x \\ \end{array}\right.\] For each of the following function restriction, determine if it is surjective.

   1. \(f|_{\mathbb{Q}}\)
   2. \(f|_{\mathbb{R}_{\geq 0}}\).
   3. \(f|_{D}\) \(D = \{ x\in \mathbb{R} \mid x \leq 2\}\).