Math Boot Camp - Class #6 · Math Boot Camp - Class #6 Alex Vickery Royal Holloway - University of...

Math Boot Camp - Class #6

Alex Vickery

Royal Holloway - University of London

25th September, 2017

Alex Vickery (Royal Holloway) Math Boot Camp - Class #6 25th September, 2017 1 / 51

Outline:Today’s Class

1 Functions and Relations:Ordered Pairs:OverviewExercises

2 Types of Function:OverviewExercises

Functions and Relations: Ordered Pairs:

Functions and Relations:Ordered Pairs:

In writing a set {a, b} we do not care about the order in which theelements a and b appear, because by definition {a, b} = {b, a}.

The pair of elements a and b are in this case an unordered pair .

When the ordering of a and b does carry a significance, we can write twodifferent ordered pairs denoted by (a, b) and (b, a).They have the property:

(a, b) 6= (b, a) unless a = b (1)

The same rules apply to sets with more than two elements. Ordered pairs,triples, etc .. collectively can be called ordered sets; they are enclosed withparentheses rather than braces.

(a, b) 6= (b, a) unless a = b (1)

When the ordering of a and b does carry a significance, we can write twodifferent ordered pairs denoted by (a, b) and (b, a).

They have the property:

(a, b) 6= (b, a) unless a = b (1)

Example (1)

To show the age and weight of each student in the class, we can formordered pairs (a,w), in which the first element indicates the age (in years)and the second element indicates the weight (in kg). Then, (19, 64) and(64,19) would obviously mean different things.

Example (2)

When we speak of the set of all contestants in the Olympic Games, theorder in which they are listed is of no consequence and we have anunordered set. But the set:(gold-medalist, silver-medalist, bronze-medalist) is an ordered triple.

Example (1)

To show the age and weight of each student in the class, we can formordered pairs (a,w), in which the first element indicates the age (in years)and the second element indicates the weight (in kg). Then, (19, 64) and(64,19) would obviously mean different things.

Example (2)

When we speak of the set of all contestants in the Olympic Games, theorder in which they are listed is of no consequence and we have anunordered set. But the set:(gold-medalist, silver-medalist, bronze-medalist) is an ordered triple.

Ordered pairs can be elements of a set.

Consider the rectangular (Cartesian) coordinate plane (on the next slide),where an x-axis and y -axis cross each other at a right angle, dividing theplane into four quadrants.

This xy plane is an infinite set of points, each represents an ordered pairwhose first element is an x value and the second element a y value.

−6 −5 −4 −3 −2 −1 1 2 3 4 5 6

(2,4) (4,4)

(Quadrant 1)(Quadrant 2)

(Quandrant 3) (Quadrant 4)

Clearly, the point labeled(4, 2) is different fromthe point (2, 4); thus,ordering is significanthere.

Suppose, from two given sets: x = {1, 2} and y = {3, 4}, we wish to formall the possible ordered pairs with the first element taken from set x andthe second taken from set y .

Result:

(1, 3), (1, 4), (2, 3), (2, 4)

This set is called the Cartesian product of the sets x and y and is denotedby: x × y (read “x cross y”).

Result:

(1, 3), (1, 4), (2, 3), (2, 4)

Result:

(1, 3), (1, 4), (2, 3), (2, 4)

Result:

(1, 3), (1, 4), (2, 3), (2, 4)

It is important to remember that while x and y are sets of numbers, theCartesian product is a a set of ordered pairs.

By enumeration, or by description , we may express the Cartesian productas:

x × y = {(1, 3), (1, 4), (2, 3), (2, 4)} (2)

x × y = {(a, b) | a ∈ x and b ∈ y} (3)

Expression (3) is the general definition of Cartesian product for any givensets x and y .

x × y = {(1, 3), (1, 4), (2, 3), (2, 4)} (2)

x × y = {(a, b) | a ∈ x and b ∈ y} (3)

x × y = {(1, 3), (1, 4), (2, 3), (2, 4)} (2)

x × y = {(a, b) | a ∈ x and b ∈ y} (3)

x × y = {(1, 3), (1, 4), (2, 3), (2, 4)} (2)

x × y = {(a, b) | a ∈ x and b ∈ y} (3)

Now, let both x and y include all the real numbers.

The resulting Cartesian product is:

x × y = {(a, b) | a ∈ R and b ∈ R} (4)

This represents the set of all ordered pairs with real valued elements.

Each ordered pair corresponds to a unique point in the Cartesiancoordinate plane, and, conversely, each point in the coordinate plane alsocorresponds to a unique ordered pair in the set x × y .

x × y = {(a, b) | a ∈ R and b ∈ R} (4)

Given this double uniqueness, a one-to-one correspondence is said to existbetween the set of ordered pairs in the Cartesian product and the set ofpoints in the rectangular coordinate plane.

The rationale for the notation: x × y is now easy to percieve; we canassociate it with the crossing of the x axis and the y axis.

A simpler way of expressing the set x × y in (4) is to write it directly as:R× R; this is commonly denoted by: R2.

Extending this, we can also define the Cartesian product of three sets x , yand z as follows:

x × y × z = {(a, b, c) | a ∈ x , b ∈ y , c ∈ z} (5)

which is a set of ordered triples.

If the sets x , y and z each consist of all the real numbers, the Cartesianproduct will correspond to the set of all points in a three dimensionalspace.

This may be denoted: R× R× R, or more simply, R3.

x × y × z = {(a, b, c) | a ∈ x , b ∈ y , c ∈ z} (5)

Functions and Relations: Overview

Functions and Relations:Overview:

Since any ordered pair associates a y value with an x value, any collectionof ordered pairs - any subset of the Cartesian product - will constitute arelation between y and x .

Given an x value, one or more y values will be specified by that relation.

Example (1)

The set: {(x , y) | y = 2x} is a set of ordered pairs including, for example,(1, 2), (0, 0) and (−1,−2). It constitutes a relation, its graphicalcounterpart is the set of points lying on the straight line: y = 2x , seen onthe next slide.

Example (1)

−3 −2 −1 1 2 3 4 5

6y = 2x

Example (2)

The set: {(x , y) | y ≤ x}, which consists of such ordered pairs as:(1, 0), (1, 1), and (1,−4), constitutes another relation. On the next slide,this set corresponds to the set of all points in the shaded area whichsatisfy the inequality: y ≤ x .

y ≤ x

Observe that, when the x value is given, it may not always be possible todetermine a unique y value from a relation.

In example 2, the three exemplary ordered pairs show that if x = 1, y cantake various values, such as: 0, 1, or −4, and in each case satisfy thestated relation.

Graphically, two or more points of a relation may fall on a single verticalline in the xy plane. This is shown on the following slide, where manypoints in the shaded area (representing the relation y ≤ x) fall on thebroken vertical line labeled: x = a.

y = xx = a

y ≤ x

As a special case, a relation may be such that for each x value there existsonly one corresponding y value. The relation in example (1) provides sucha case.

When we have this type of relation, y is said to be a function of x . This isdenoted by: y = f (x) (read as: “y equals f of x”).

A function is therefore a set of ordered pairs with the property that any xvalue uniquely determines a y value. It should be clear that a functionmust be a relation, but a relation may not be a function.

Although the definition of a function stipulates a unique y for each x , theconverse is not required.

More than one x value may be legitimately associated with the same yvalue.

This possibility is demonstrated on the next slide, where the values x1 andx2 in the x set are both associated with the same value (y0) in the y setby the function y = f (x).

y = f (x)

x1 x20

A function is also called a mapping , or transformation; both wordsconnote the action of associating one thing with another.

In the statement y = f (x), the functional notation f may thus beinterpreted to mean a rule by which the set x is “mapped”(“transformed”) to the set y . We may write:

f : x → y

where the arrow indicates mapping, and the letter f symbolically specifiesa rule of mapping.

f : x → y

Since f represents a particular rule of mapping, a different functionalnotation must be employed to denote another function appearing in thesame model.

Example

Two variables y and z may both be functions of x .If one is written as: y = f (x), the other should be written: z = g(x).

Since f represents a particular rule of mapping, a different functionalnotation must be employed to denote another function appearing in thesame model.

Example

Two variables y and z may both be functions of x .If one is written as: y = f (x), the other should be written: z = g(x).

In the function y = f (x), x is referred to as the argument of the function,and y is called the value of the function.

Alternatively, we can refer to x as the independant variable and to y asthe dependent variable.

In the function y = f (x), x is referred to as the argument of the function,and y is called the value of the function.

Alternatively, we can refer to x as the independant variable and to y asthe dependent variable.

The set of all permissible values that x can take in a given context isknown as the domain of the function, which may be a subset of the realnumbers.

The y value into which an x value is mapped is called the image of that xvalue.

The set of all images is called the range of the function, which is the setof all values that the y variable can take.

Thus the domain pertains to the independent variable x , and the rangehas to do with the dependent variable y .

x1 x2 y1 y2

Domain Range

As illustrated here, wemay regard the functionf as a rule for mappingeach point on some linesegment (the domain)into some point onanother line segment (therange).

(x1, y1)

(x2, y2)

By placing the domain onthe x axis and the rangeon the y axis, weimmediately obtain thefamiliar two-dimensionalgraph, in which theassociation between the xand y values is specifiedby a set of ordered pairs:(x1, y1), (x2, y2).

Example

The total cost, C of a firm per day is a function of its daily output, Q:C = 150 + 7Q. The firm has a capacity limit of 100 units of output perday. What are the domain and the range of the cost function?

Q can vary only between 0 and 100, the domain is the set of values0 ≤ Q ≤ 100; more formally:

Domain = {Q | 0 ≤ Q ≤ 100}

Since the function plots a straight line, with the minimum C value at 150(when Q =0) and the maximum C value at 850 (when Q = 100), wehave:

Range = {C | 150 ≤ C ≤ 850}

Example

Domain = {Q | 0 ≤ Q ≤ 100}

Range = {C | 150 ≤ C ≤ 850}

Example

Domain = {Q | 0 ≤ Q ≤ 100}

Range = {C | 150 ≤ C ≤ 850}

Example

Domain = {Q | 0 ≤ Q ≤ 100}

Range = {C | 150 ≤ C ≤ 850}

Example

Domain = {Q | 0 ≤ Q ≤ 100}

Range = {C | 150 ≤ C ≤ 850}

Functions and Relations: Exercises

Functions and Relations:Exercises:

1.) Given S1 = {3, 6, 9}, S2 = {a, b}, S3 = {m, n}, find the Cartesianproducts:

S1 × S2(i) S2 × S3(ii) S3 × S1(iii)

2.) Do any of the following, drawn in the rectangular coordinate plane,represent a function?

A circle(i) A rectangle(ii) A triangle(iii)

3.) If the domain of the function y = 5 + 3x is the set {x | 1 ≤ x ≤ 9},find the range of the function and express it as a set.

4.) For the function y = −x2, if the domain is the set of all nonnegativereal numbers, what will its range be?

Functions and Relations:Exercises:

1.) Given S1 = {3, 6, 9}, S2 = {a, b}, S3 = {m, n}, find the Cartesianproducts:

S1 × S2(i) S2 × S3(ii) S3 × S1(iii)

2.) Do any of the following, drawn in the rectangular coordinate plane,represent a function?

A circle(i) A rectangle(ii) A triangle(iii)

3.) If the domain of the function y = 5 + 3x is the set {x | 1 ≤ x ≤ 9},find the range of the function and express it as a set.

4.) For the function y = −x2, if the domain is the set of all nonnegativereal numbers, what will its range be?

Types of Function: Overview

Types of Function:Overview:

The expression y = f (x) is a general statement to the effect that themapping is possible, but the actual rule of the mapping is not therebymade explicit.

Let us now consider several specific types of function, each representing adifferent rule of mapping.

The expression y = f (x) is a general statement to the effect that themapping is possible, but the actual rule of the mapping is not therebymade explicit.

Let us now consider several specific types of function, each representing adifferent rule of mapping.

Constant Functions:

A function whose range consists of only one element is called aconstant function. As an example:

y = f (x) = 7

which is alternatively expressable as y = 7 or f (x) = 7, whose value staysthe same regardless of the value of x .

In the coordinate plane, such a function will appear as a horizontalstraight line.

Constant Functions:

y = f (x) = 7

Constant Functions:

y = f (x) = 7

Constant Functions:

y = f (x) = 7

Constant Functions:

y = f (x) = 7

Polynomial Functions:

The constant function is a “degenerate” case of what are known aspolynomial functions. A polynomial function of a single variable x has thegeneral form:

y = a0 + a1x + a2x2 + · · ·+ anx

in which each term contains a coefficient as well as a nonnegative integerpower of the variable x .

y = a0 + a1x + a2x2 + · · ·+ anx

Depending on the value of the integer n ()which specifies the highestpower of x), we have several subclasses of polynomial function:

Symbol Form Name

n = 0 : y = a0 [constant function]

n = 1 : y = a0 + a1x [linear function]

n = 2 : y = a0 + a1x + a2x2 [constant function]

n = 3 : y = a0 + a1x + a22x + a3x3 [cubic function]

Depending on the value of the integer n ()which specifies the highestpower of x), we have several subclasses of polynomial function:

Symbol Form Name

n = 0 : y = a0 [constant function]

n = 1 : y = a0 + a1x [linear function]

n = 2 : y = a0 + a1x + a2x2 [constant function]

n = 3 : y = a0 + a1x + a22x + a3x3 [cubic function]

The subscript indicators of the powers are called exponents. The highestpower involved (the value of n), is often called the degree of thepolynomial function.

For example, a quadratic function is a second-degree polynomial, and acubic function is a third-degree polynomial.

The order in which the terms appear to the right of the equals sign isinconsequential.

When plotted in the coordinate plane, a linear function will appear as astraight line (see next slide).

When x = 0, the linear function yields y = a0 (the ordered pair (0, a0) ison the line).

This gives us the vertical intercept, because it is at this point that thevertical axis intersects the line.

The other coefficient, a1, measures the slope of the line. This means thata unit increase in x will result in an increment in y of the amount a1.

Types of Functions:Overview:

Slope = a1

Lineary = a0 + a1x

Here the case wherea1 > 0 is illustrated,involving a positive slopeand an upward slopingline; if a1 < 0, the linewould be downwardsloping.

A quadratic function plots a parabola - roughly, a curve with a single builtin bump or wiggle.

The graph on the next slide implies a negative value for a2; in the case ofa2 > 0, the curve will “open” the other way, displaying a valley rather thana hill.

A quadratic function plots a parabola - roughly, a curve with a single builtin bump or wiggle.

The graph on the next slide implies a negative value for a2; in the case ofa2 > 0, the curve will “open” the other way, displaying a valley rather thana hill.

(Case of a2 < 0)

Quadratic

y = a0 + a1x + a2x2

Rational Functions:

A function such as:

y =x − 1

x2 + 2x − 4

in which y is expressed as a ratio of two polynomials in the variable x , isknown as a rational function.

According to the definition, any polynomial function itself must be arational function, because it can always be expressed as a ratio to 1, and 1is a constant function.

Rational Functions:

A function such as:

y =x − 1

x2 + 2x − 4

Rational Functions:

A function such as:

y =x − 1

x2 + 2x − 4

Rational Functions:

A function such as:

y =x − 1

x2 + 2x − 4

A special rational function that has interesting applications in economics isthe function:

xor xy = a

which plots a rectangular hyperbola, (see next slide).

Since the product of the two variables is always a fixed constant in thiscase, this function may be used to represent the demand curve - with priceP and quantity Q on the two axes.

xor xy = a

(a > 0)

Rectangular-hyperbolicy = a

The rectangular hyperbola drawn from xy = a never meets the axes, evenif extended indefinitely upward and to the right.

The curve approaches the axes asymptotically : as y becomes very large,the curve will approach the y axis but never actually reach it, and similarlyfor the x axis.

The axes constitute the asymptotes of this function.

Nonalgebraic Functions:

Any function expressed in terms of polynomials and/or roots ofpolynomials is an algebraic function.

However, exponential functions such as y = bx , in which the independentvariable appears in the exponent, are nonalgebraic .

An example is given on the next slide.

(b > 1)

Exponential

y = bx

The closely related logarithmic functions, such as y = logb x , are alsononalgebraic.

Nonalgebraic functions are also known by the more esoteric name oftranscendental functions.

(b > 1)

Logarithmic

y = logb x

Types of Function: Exercises

Types of Functions:Exercises:

1.) Graph the functions:

y = 16 + 2x(i) y = 8− 2x(ii) y = 2x + 12(iii)

2.) Graph the functions:

y = −x2 + 5x −2(i) y = x2 + 5x − 2(ii)

3.) Graph the function y = 36/x , assuming the x and y can only takepositive values. Next, suppose that both variables can take negativevalues as well; how must the graph be changed to reflect this?

Math Boot Camp - Class #6 · Math Boot Camp - Class #6 Alex Vickery Royal Holloway - University of...

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