Chapter 1.3 Practice

Questions

Question 1

  1. Let \(X=\{a,0,2\}\) and \(Y=\{c,d,4\}\). Write out \(X\times Y\).

Question 2

  1. Take \(Y=\{c,d,e,f\}\) and \(r=\{(2,c),(6,e),(11,e),(16,c)\}.\) View \(r\) as a subset of \(\mathbb{N}\times Y\).

    1. What is the natural domain of \(r\)?
    2. What is the co-domain of \(r\)?
    3. What is the domain of \(r\) (denoted by \(\mathcal{D}(r)\))?
    4. What is the range of \(r\) (denoted by \(\mathcal{R}(r)\))?

Question 3

  1. For each relation, determine if it is a function or not.

    1. The relation \(r=\{(\pi,\pi),(-\pi,\pi),(5,-1),(10,6)\}.\)
    2. The relation \(r=\{(3,\star),(2,\square),(0,0),(2,1)\}.\)
    3. The relation in \(\mathbb{R}\times \mathbb{R}\) whose graph is given below

    1. The relation in \(\mathbb{R}\times \mathbb{R}\) whose graph is given below.

Question 4

  1. For each function given, state the domain and the range.

    1. The function \(f\colon \mathbb{R}\times \mathbb{R}\) whose graph is given below.

    1. The function \(f\colon\mathbb{R}\times \mathbb{R}\) whose graph is given below.

    1. The function \(f\colon\mathbb{R}\times \mathbb{R}\) whose graph is given below.

Question 5

  1. For each function given, find all intervals where \(f\) is non-negative, positive, non-positive, and negative.
    1. The function \(f\) whose graph is given below.

    1. The function \(f\) whose graph is given below.

    1. The function \(f\) whose graph is given below.

Question 6

  1. For each function given, find the largest intervals on which \(f\) is strictly increasing and strictly decreasing.
    1. The function \(f\) whose graph is given below.

    1. The function \(f\) whose graph is given below.

    1. The function \(f\) whose graph is given below.

    1. The function \(f\) whose graph is given below.

Question 7

  1. For each function given, find all extremal values of \(f\) in \([-4,3]\)
    1. The function \(f\) whose graph is given below.

    1. The function \(f\) whose graph is given below.

Answers

Question 1

  1. \(X\times Y=\{(a,c),(a,d),(a,4),(0,c),(0,d),(0,4),(2,c),(2,d),(2,4)\}.\)

Question 2

    1. \(\mathbb{N}\)
    2. \(Y\)
    3. \(\mathcal{D}(r)=\{2,6,11,16\}\)
    4. \(\mathcal{R}(r)=\{c,e\}\)

Question 3

    1. A function
    2. Not a function
    3. Not a function
    4. A function

Question 4

    1. \(\mathcal{D}=[-4,-2]\cup(-1,3]\) and \(\mathcal{R}=[-2,4]\)
    2. \(\mathcal{D}=[-2,2]\) and \(\mathcal{R}=[-4,0]\cup[1,3)\)
    3. \(\mathcal{D}=[-6,-4)\cup(-4,0)\cup(0,1]\) and \(\mathcal{R}=[-2,0)\cup(0,1]\)

Question 5

    1. non-negative on \([-1,1]\), positive on \((-1,1)\), non-positive on \([-2,-1]\cup[1,4]\), negative on \([-2,-1)\cup(1,4]\).
    2. non-negative on \([-2,0]\cup\{4\}\), positive on \((-2,0)\), non-positive on \((0,4]\), negative on \((0,4)\).
    3. non-negative on \(\{-1\}\), positive on \(\{\}\), non-positive on \((-\infty,3]\), negative on \((-\infty,-1)\cup(-1,3]\).

Question 6

    1. increasing on \((-2,0)\); decreasing on \((0,4)\)
    2. increasing on \((-2,0)\); decreasing on \((0,4)\)
    3. increasing on \((-\infty,-1)\); decreasing on \((-1,3)\)
    4. increasing on \((-3,1)\); decreasing on \((-4,-3),(1,\infty)\)

Question 7

    1. local max at \(x=-4\), \(x=1\); local min at \(x=-3\), \(x=3\); global max at \(x=1\); global min at \(x=-3\).
    2. local max at \(x=2\); local min at \(x=-4\), \(x=3\); global max at \(x=2\); global min at \(x=-4\).

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