1.1.1: exercise temp

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1.1: Sets of Real Numbers and the Cartesian Coordinate Plane

Exercise $\PageIndex{1.1.1}$:

Fill in the chart below:

$1.1 Exercise table 1.PNG$

In Exercises 1.1.2 - 1.1.7, find the indicated intersection or union and simplify if possible. Express your answers in interval notation.

$ (-1,5]\cap[0,8)$

Exercise $\PageIndex{1.1.3}$:

$ (-1,1)\ \cup\ [0,6]$

Exercise $\PageIndex{1.1.4}$:

$ (-\infty,4]\cap(0,\infty)$

Exercise $\PageIndex{1.1.5}$:

$ (-\infty,0)\cap[1,5] $

Exercise $\PageIndex{1.1.6}$:

$ \( (-\infty,0)\cup[1,5] $\)

$ (-\infty,5]\cap[5,8)$

In Exercises 1.1.8 - 1.1.19, write the set using interval notation.

$ \{x\,|\, x \neq 0, \ 2 \}$

Exercise $\PageIndex{1.1.9}$:

$ \{x|x\neq-1\}$

Exercise $\PageIndex{1.1.10}$:

$ \{x|x\neq-3,\ 4\}$

Exercise $\PageIndex{1.1.11}$:

$ \{x|x\neq0,\ 2\}$

Exercise $\PageIndex{1.1.12}$:

$ \{x\,|\, x \neq 2,\ -2 \}$

Exercise $\PageIndex{1.1.13}$:

$ \{x|x\neq0,\ \pm4\}$

Exercise $\PageIndex{1.1.14}$:

$ \{x|x\leq-1\ \text{or}\ x\geq 1\}$

Exercise $\PageIndex{1.1.15}$:

$ \{x|x<3\ \text{or}\ x\geq 2\}$

Exercise $\PageIndex{1.1.16}$:

$ \{x|x\leq-3\ \text{or}\ x>0\}$

Exercise $\PageIndex{1.1.17}$:

$ \{x|x\leq5\ \text{or}\ x=6\}$

Exercise $\PageIndex{1.1.18}$:

$ \{x|x>2\ \text{or}\ x=\pm1\}$

$ \{x|-3<x<3\ \text{or}\ x=4\}$

Exercise $\PageIndex{1.1.20}$:

Plot and label the points $A(-3, -7)$, $B(1.3, -2)$, $C(\pi, \sqrt{10})$, $D(0, 8)$, $E(-5.5, 0)$, $F(-8, 4)$, $G(9.2, -7.8)$ and $H(7, 5)$ in the Cartesian Coordinate Plane given below.

$Cartesian Coord 1.20.PNG$

Exercise $\PageIndex{1.1.21}$:

For each point given in Exercise 20 above:

Identify the quadrant or axis in/on which the point lies.
Find the point symmetric to the given point about the $x$-axis.
Find the point symmetric to the given point about the $y$-axis.
Find the point symmetric to the given point about the origin.

In Exercises 1.1.22 - 1.1.29, find the distance $d$ between the points and the midpoint $M$ of the line segment which connects them.

Exercise $\PageIndex{1.1.22}$:

$ (1,\ 2),\ (3,\ 5)$

Exercise $\PageIndex{1.1.23}$:

$ (3,\ 10),\ (1,\ 2)$

Exercise $\PageIndex{1.1.24}$:

$ (\frac{1}{2},\ 4),\ (\frac{3}{2},\ -1)$

Exercise $\PageIndex{1.1.25}$:

$ (-\frac{2}{3},\ \frac{3}{2}),\ (\frac{7}{3},\ 2)$

Exercise $\PageIndex{1.1.26}$:

$ (-\frac{24}{5},\ \frac{6}{5}),\ (-\frac{11}{5},\ -\frac{19}{5})$

Exercise $\PageIndex{1.1.27}$:

$ (\sqrt{2},\ \sqrt{3}),\ (-\sqrt{8},-\sqrt{12})$

Exercise $\PageIndex{1.1.28}$:

$ (2\sqrt{45},\ \sqrt{12}),\ (\sqrt{20},\sqrt{27})$

Exercise $\PageIndex{1.1.29}$:

$(0,\ 0),\ (x,\ y) $

Exercise $\PageIndex{1.1.30}$:

Find all of the points of the form $(x,\ -1)$ which are 4 units from the point $(3,\ 2)$.

Exercise $\PageIndex{1.1.31}$:

Find all of the points on the $ y$-axis which are 5 units from the point $(-5,\ 3)$.

Exercise $\PageIndex{1.1.32}$:

Find all of the points on the $ x$-axis which are 2 units from the point $(-1,\ 1)$.

Exercise $\PageIndex{1.1.33}$:

Find all of the points of the form $(x,\ -x)$ which are 1 unit from the origin.

Exercise $\PageIndex{1.1.34}$:

Let's assume for a moment that we are standing at the origin and the positive $ y$-axis points due North while the positive $ x$-axis points due East. Our Sasquatch-o-meter tells us that Sasquatch is 3 miles West and 4 miles South of our current position. What are the coordinates of his position? How far away is he from us? If he runs 7 miles due East what would his new position be?

Exercise $\PageIndex{1.1.35}$:

Add text here. For the automatic number to work, you need to add the "AutoNum" template (preferably at the end) to the page.

Exercise $\PageIndex{A}$:

The points are arranged vertically. (Hint: Use $P(a,\ y_0)$ and Q(a, y_1).)

Exercise $\PageIndex{B}$:

The points are arranged horizontally. (Hint: Use $P(x_0,\ b)$ and $Q(x_1,\ b)$.)

Exercise $\PageIndex{C}$:

The points are actually the same point. (You shouldn't need a hint for this one.)

Exercise $\PageIndex{1.1.36}$:

Verify the Midpoint Formula by showing the distance between $P(x_1,\ y_1)$ and $ M$ and the distance between $ M$ and $ Q(x_2,\ y2)$ are both half of the distance between $P$ and $Q$.

Exercise $\PageIndex{1.1.37}$:

Show that the points $ A$, $ B$ and $ C$ below are the vertices of a right triangle.

Exercise $\PageIndex{A}$:

$ A(-3,2),\ B(-6,4)$, and $C(1,8)$

Exercise $\PageIndex{B}$:

$ A(-3,\ 1),\ B(4,\ 0)$, and $C(0,\ 3)$

Exercise $\PageIndex{1.1.38}$:

Find a point $D(x,\ y)$ such that the points $A(-3,\ 1)$, $B(4,\ 0)$, $C(0,\ 3)$ and $ D$ are the corners of a square. Justify your answer.

Exercise $\PageIndex{1.1.39}$:

Discuss with your classmates how many numbers are in the interval $ (0,\ 1)$.

Exercise $\PageIndex{1.1.40}$:

The world is not at. Thus the Cartesian Plane cannot possibly be the end of the story. Discuss with your classmates how you would extend Cartesian Coordinates to represent the three dimensional world. What would the Distance and Midpoint formulas look like, assuming those concepts make sense at all?

1.2: Relations

In Exercises 1 - 20, graph the given relation.

Exercise $\PageIndex{1}$:

$ \{(-3, 9), (-2, 4), (-1, 1), (0, 0), (1, 1), (2, 4), (3, 9)\} $

Exercise $\PageIndex{2}$:

$ \{(-2, 0), (-1, 1), (-1,-1), (0, 2), (0,-2), (1, 3), (1,-3)\} $

Exercise $\PageIndex{3}$:

$\{(m,\ 2m) |m = 0;-1,\pm2\}$

Exercise $\PageIndex{4}$:

$ \{(\frac{6}{k},\ k)\ |k = \pm1,\pm2,\pm3,\pm4,\pm5,\pm6\}$

Exercise $\PageIndex{5}$:

$ \{(n,4-n^2)\ |n=0,\pm 1,\pm 2\} $

Exercise $\PageIndex{6}$:

$ \{(\sqrt{j},j)\ |j=0,1,4,9\} $

Exercise $\PageIndex{7}$:

$ \{(x,-2)\ |x>-4\} $

Exercise $\PageIndex{8}$:

$ \{(x,3)\ |x\leq4\} $

Exercise $\PageIndex{9}$:

$ \{(-1,y)\ |y>1\} $

Exercise $\PageIndex{10}$:

$ \{(2,y)\ |y\leq5\} $

Exercise $\PageIndex{11}$:

$ \{(-2,y)\ |-3<y\leq4\} $

Exercise $\PageIndex{12}$:

$ \{(3,y)\ |-4\leq y<3\} $

Exercise $\PageIndex{13}$:

$ \{(x,2)|\ -2\leq x<3\} $

Exercise $\PageIndex{14}$:

$ \{(x,-3)|\ -4<x\leq3\} $

Exercise $\PageIndex{15}$:

$ \{(x,y)\ |x>-2\} $

Exercise $\PageIndex{16}$:

$ \{(x,y)\ |x\leq3\} $

Exercise $\PageIndex{17}$:

$ \{(x,y)\ |y<4\} $

Exercise $\PageIndex{18}$:

$ \{(x,y)\ |x\leq3,y<2\} $

Exercise $\PageIndex{19}$:

$ \{(x,y)\ |x>0,y<4\} $

Exercise $\PageIndex{20}$:

$ \{(x,y)\ |-\sqrt{2}\leq x\leq \frac{2}{3},\pi <y\leq \frac{9}{2}\} $

In Exercises 21 - 30, describe the given relation using either the roster or set-builder method.

Exercise $\PageIndex{21}$:

$1.2 Relation A.PNG$

Exercise $\PageIndex{22}$:

$1.2 Relation B.PNG$

Exercise $\PageIndex{23}$:

$1.2 Relation C.PNG$

Exercise $\PageIndex{24}$:

$1.2 Relation D.PNG$

Exercise $\PageIndex{25}$:

$1.2 Relation E.PNG$

Exercise $\PageIndex{26}$:

$1.2 Relation F.PNG$

Exercise $\PageIndex{27}$:

$1.2 Relation G.PNG$

Exercise $\PageIndex{28}$:

$1.2 Relation H.PNG$

Exercise $\PageIndex{29}$:

$1.2 Relation I.PNG$

Exercise $\PageIndex{30}$:

$1.2 Relation J.PNG$

In Exercises 31 - 36, graph the given line.

Exercise $\PageIndex{31}$:

$x=-2$

Exercise $\PageIndex{32}$:

$ x=3$

Exercise $\PageIndex{33}$:

$ y=3$

Exercise $\PageIndex{34}$:

$ y=-2$

Exercise $\PageIndex{35}$:

$ x=0$

Exercise $\PageIndex{36}$:

$ y=0$

Some relations are fairly easy to describe in words or with the roster method but are rather diffcult, if not impossible, to graph. Discuss with your classmates how you might graph the relations given in Exercises 37 - 40. Please note that in the notation below we are using the ellipsis, . . . , to denote that the list does not end, but rather, continues to follow the established pattern indenitely. For the relations in Exercises 37 and 38, give two examples of points which belong to the relation and two points which do not belong to the relation.

Exercise $\PageIndex{37}$:

$ \{ (x,y)\ |x\ \text{is an odd integer, and } y\ \text{is an even integer.}\}$

Exercise $\PageIndex{38}$:

$ \{(x,\ 1)\ |x\text{ is an irrational number }\}$

Exercise $\PageIndex{39}$:

$ \{ (1,\ 0),\ (2,\ 1),\ (4,\ 2),\ (8,\ 3),\ (16,\ 4),\ (32,\ 5),...\} $

Exercise $\PageIndex{40}$:

$ \{...,(-3,\ 9),(-2,\ 4),(-1,\ 1),(0,\ 0),(1,\ 1),(2,\ 4),(3,\ 9)...\} $

For each equation given in Exercises 41 - 52:

Find the x- and y-intercept(s) of the graph, if any exist.
Follow the procedure in Example 1.2.3 to create a table of sample points on the graph of the
equation.
Plot the sample points and create a rough sketch of the graph of the equation.
Test for symmetry. If the equation appears to fail any of the symmetry tests, find a point on the graph of the equation whose reflection fails to be on the graph as was done at the end of Example 1.2.4

Exercise $\PageIndex{41}$:

$ y=x^{2}+1$

Exercise $\PageIndex{42}$:

$ y=x^{2}\ -\ 2x\ -\ 8$

Exercise $\PageIndex{43}$:

$ y=x^{3}-x$

Exercise $\PageIndex{44}$:

$ y=\frac{x^{3}}{4}-3x$

Exercise $\PageIndex{45}$:

$ y=\sqrt{x-2}$

Exercise $\PageIndex{46}$:

$ y=2\sqrt{x+4}-2$

Exercise $\PageIndex{47}$:

$ 3x\ -\ y=7$

Exercise $\PageIndex{48}$:

$ 3x-2y=10$

Exercise $\PageIndex{49}$:

$ (x+2)^{2}+y^{2}=16$

Exercise $\PageIndex{50}$:

$ x^{2}-y^{2}=1$

Exercise $\PageIndex{51}$:

$ y=x^{2}+1$

Exercise $\PageIndex{52}$:

$ x^{3}y=-4$

The procedures which we have outlined in the Examples of this section and used in Exercises 41 - 52 all rely on the fact that the equations were "well-behaved". Not everything in Mathematics is quite so tame, as the following equations will show you. Discuss with your classmates how you might approach graphing the equations given in Exercises 53 - 56. What difficulties arise when trying to apply the various tests and procedures given in this section? For more information, including pictures of the curves, each curve name is a link to its page at www.Wikipedia.org. For a much
longer list of fascinating curves, click here.

Exercise $\PageIndex{53}$:

Folium of Descartes:

$ x^{3}+y^{3}-3xy\ =\ 0$

Exercise $\PageIndex{54}$:

Kampyle of Eudoxus:

$ x^{4}\ =\ x^{2}+y^{2}$

Exercise $\PageIndex{55}$:

Tschirnhausen cubic:

$ y^{2}\ =\ x^{3}+3x^{2}$

Exercise $\PageIndex{56}$:

Cooked egg:

$ (x^{2}+y^{2})^{2}\ =\ x^{3}+y^{3}$

Exercise $\PageIndex{57}$:

With the help of your classmates, nd examples of equations whose graphs possess

symmetry about the $x$-axis only
symmetry about the $y$-axis only
symmetry about the origin only
symmetry about the $x$-axis, $y$-axis, and origin

Can you nd an example of an equation whose graph possesses exactly two of the symmetries
listed above? Why or why not?

1.3 Introduction to Functions

In Exercises 1 - 12, determine whether or not the relation represents y as a function of x. Find the domain and range of those relations which are functions.

Exercise $\PageIndex{1}$:

$ \{(-3,9),\ (-2,4),\ (-1,1),\ (0,0),\ (1,1),\ (1,1),\ (1,1),\ (2,4),\ (3,9)\}$

Exercise $\PageIndex{2}$:

$ \{(-3,0),\ (1,6),\ (2,-3),\ (4,2),\ (-5,6),\ (4,-9),\ (6,2)\}$

Exercise $\PageIndex{3}$:

$ \{(-3,0),\ (-7,6),\ (5,5),\ (6,4),\ (4,-9),\ (3,0)\}$

Exercise $\PageIndex{4}$:

$ \{(1,2),\ (4,4),\ (9,6),\ (16,8),\ (25,10),\ (36,12),...\}$

Exercise $\PageIndex{5}$:

$ \{(x,y)\ |x \text{ is an odd integer, and }y\text{ is an even integer}\}$

Exercise $\PageIndex{6}$:

$ \{(x,1)\ |x \text{ is an irrational number}\}$

Exercise $\PageIndex{7}$:

$ \{(1,0),\ (2,1),\ (4,2),\ (8,3),\ (16,4),\ (32,5),...\}$

Exercise $\PageIndex{8}$:

$..., \{(-3,9),\ (-2,4),\ (-1,1),\ (0,0),\ (1,1),\ (2,4),\ (3,9)...\}$

Exercise $\PageIndex{9}$:

$ \{ (-2,y)|-3<y<4\}$

Exercise $\PageIndex{10}$:

$ \{ (x,3)|-2\leq x <4\}$

Exercise $\PageIndex{11}$:

$ \{(x,x^{2})\ |x \text{ is an real number}\}$

Exercise $\PageIndex{12}$:

$ \{(x^{2},x)\ |x \text{ is an real number}\}$

In Exercises 13 - 32, determine whether or not the relation represents y as a function of x. Find the domain and range of those relations which are functions.

Exercise $\PageIndex{13}$:

$1.3 Func 1.PNG$

Exercise $\PageIndex{14}$:

$1.3 Func 2.PNG$

Exercise $\PageIndex{15}$:

$1.3 Func 3.PNG$

Exercise $\PageIndex{16}$:

$1.3 Func 4.PNG$

Exercise $\PageIndex{17}$:

$1.3 Func 5.PNG$

Exercise $\PageIndex{18}$:

$1.3 Func 6.PNG$

Exercise $\PageIndex{19}$:

$1.3 Func 7.PNG$

Exercise $\PageIndex{20}$:

$1.3 Func 8.PNG$

Exercise $\PageIndex{21}$:

$1.3 Func 9.PNG$

Exercise $\PageIndex{22}$:

$1.3 Func 10.PNG$

Exercise $\PageIndex{23}$:

Exercise $\PageIndex{24}$:

$1.3 Func 12.PNG$

Exercise $\PageIndex{25}$:

$1.3 Func 13.PNG$

Exercise $\PageIndex{26}$:

$1.3 Func 14.PNG$

Exercise $\PageIndex{27}$:

$1.3 Func 15.PNG$

Exercise $\PageIndex{28}$:

Exercise $\PageIndex{29}$:

$1.3 Func 17.PNG$

Exercise $\PageIndex{30}$:

$1.3 Func 18.PNG$

Exercise $\PageIndex{31}$:

$1.3 Func 19.PNG$

Exercise $\PageIndex{32}$:

$1.3 Func 20.PNG$

In Exercises 33 - 47, determine whether or not the equation represents $ y$ as a function of $ x$.

Exercise $\PageIndex{33}$:

$ y\ =\ x^{3}\ -\ x$

Exercise $\PageIndex{34}$:

$ y\ =\ \sqrt{x-2}$

Exercise $\PageIndex{35}$:

$ x^{3}y\ =\ -4$

Exercise $\PageIndex{36}$:

$ x^{2}\ -\ y^{2}\ =\ 1$

Exercise $\PageIndex{37}$:

$ y=\frac{x}{x^{2}-9}$

Exercise $\PageIndex{38}$:

$ x\ =\ -6$

Exercise $\PageIndex{39}$:

$ x\ =\ y^{2}\ +\ 4$

Exercise $\PageIndex{40}$:

$ y\ =\ x^{2}\ +\ 4$

Exercise $\PageIndex{41}$:

$ x^{2}\ +\ y^{2}\ =\ 4$

Exercise $\PageIndex{42}$:

$ y\ =\ \sqrt{4-x^{2}}$

Exercise $\PageIndex{43}$:

$ x^{2}\ -\ y^{2}\ =\ 4$

Exercise $\PageIndex{44}$:

$ x^{3}\ +\ y^{3}\ =\ 4$

Exercise $\PageIndex{45}$:

$ 2x\ +\ 3y\ =\ 4$

Exercise $\PageIndex{46}$:

$ 2xy\ =\ 4$

Exercise $\PageIndex{47}$:

$ x^{2}\ =\ y^{2}$

Exercise $\PageIndex{48}$:

Explain why the population $ P$ of Sasquatch in a given area is a function of time $ t$. What would be the range of this function?

Exercise $\PageIndex{49}$:

Explain why the relation between your classmates and their email addresses may not be a function. What about phone numbers and Social Security Numbers?

The process given in Example 1.3.5 for determining whether an equation of a relation represents $ y$ as a function of $ x$ breaks down if we cannot solve the equation for $ y$ in terms of $ x$. However, that does not prevent us from proving that an equation fails to represent $ y$ as a function of $ x$. What we really need is two points with the same $ x$-coordinate and different $ y$-coordinates which both satisfy the equation so that the graph of the relation would fail the Vertical Line Test 1.1. Discuss with your classmates how you might find such points for the relations given in Exercises 50 - 53.

Exercise $\PageIndex{50}$:

$ x^{3}\ +\ y^{3}\ -3xy\ =\ 0$

Exercise $\PageIndex{51}$:

$ x^{4}\ =\ x^{2}\ +\ y^{2}$

Exercise $\PageIndex{52}$:

$ y^{2}\ =\ x^{3}\ +\ 3x^{2}$

Exercise $\PageIndex{53}$:

$ (x^{2}+y^{2})^{2}\ =\ x^{3}\ +\ y^{3}$