Calculus by Beamer

BUSCALC

YvetteFajardo-Lim

FUNCTIONSAND THEIRGRAPHSDefinitions andExamples

Linear Functions

QuadraticFunctions

Rational Functions

Inverse Functions

ExponentialFunctions

LogarithmicFunctions

BUSCALC LECTURE NOTESCHAPTER 1

Yvette Fajardo-Lim

Mathematics DepartmentDe La Salle University - Manila

BUSCALC

YvetteFajardo-Lim


Linear Functions

QuadraticFunctions

Rational Functions

Inverse Functions



Outline

1 FUNCTIONS AND THEIR GRAPHSDefinitions and ExamplesLinear FunctionsQuadratic FunctionsRational FunctionsInverse FunctionsExponential FunctionsLogarithmic Functions

BUSCALC

YvetteFajardo-Lim


Linear Functions

QuadraticFunctions

Rational Functions

Inverse Functions



Functions

DefinitionA function, denoted by f , is a rule that assigns to eachobject x in a set X exactly one object f (x) in a set Y . Theelement f (x) in Y is called the image of x under f . The setX is called the domain of the function and Y its codomain.The set of assigned objects in Y is called the range of thefunction f , i.e., the range of f is the set {f (x)|x ∈ X} ⊆ Y.

BUSCALC

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Linear Functions

QuadraticFunctions

Rational Functions

Inverse Functions



Functions

ExampleConsider the equation y = 2x; this defines a function f forwhich the domain X is the set of all real numbers and therange of f is the set of all even numbers. Hence, f (x) = 2xis the image of x under f .

BUSCALC

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Linear Functions

QuadraticFunctions

Rational Functions

Inverse Functions



Functions

Since the value of the variable y in y = f (x) alwaysdepends on the choice of x , we say that y is the dependentvariable and since x is independent of y , x is called theindependent variable.

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Linear Functions

QuadraticFunctions

Rational Functions

Inverse Functions



Composition of Functions

DefinitionGiven the functions f (x) and g(x),

1 The composition of f ◦ g is defined by(f ◦ g)(x) = f (g(x))

2 The composition of g ◦ f is defined by(g ◦ f )(x) = g(f (x))

BUSCALC

YvetteFajardo-Lim


Linear Functions

QuadraticFunctions

Rational Functions

Inverse Functions



Composition of Functions

Example

Given f (x) = 2x2 − x + 3,g(x) = x + 2 and h(x) =√

x − 2

1 (f ◦ g)(x) = 2x2 + 7x + 92 (g ◦ f )(x) = 2x2 − x + 53 (f ◦ h)(x) = 2x −

√x − 2− 1

4 (h ◦ g)(x) =√

x

BUSCALC

YvetteFajardo-Lim


Linear Functions

QuadraticFunctions

Rational Functions

Inverse Functions



Domain and Range of Functions

When defining a function, the domain must be given eitherimplicitly or explicitly. Unless otherwise specified, thedomain of the function is the set of all real numbers forwhich f (x) is defined. Most often, to determine the domainof a function, all values of x that result in division by 0 ortaking the root of a negative number are excluded.

BUSCALC

YvetteFajardo-Lim


Linear Functions

QuadraticFunctions

Rational Functions

Inverse Functions



Domain and Range of Functions

Example

The domain of f (x) =√

x + 4 is the set of all real numbers xsuch that x ≥ −4. The range of f is the set of allnonnegative real numbers.

Example

The domain of f (x) =2

x − 2is the set of all real numbers x

such that x 6= 2. The range of f is the set of all real numberssuch that y 6= 0.

BUSCALC

YvetteFajardo-Lim


Linear Functions

QuadraticFunctions

Rational Functions

Inverse Functions



Piecewise-defined Functions

Functions can also be defined using different rules ondisjoint subset of its domain. A function defined this way iscalled a piecewise-defined function.

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Linear Functions

QuadraticFunctions

Rational Functions

Inverse Functions



Piecewise-defined Functions

Example

Given f (x) ={

2x + 8 if x < −4√x + 4 if x ≥ −4

f (−6) = −4 and f (5) = 3.

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Linear Functions

QuadraticFunctions

Rational Functions

Inverse Functions



The Graph of a Function

DefinitionThe graph of a function f is the set of all points{(x , y) : y = f (x)} on the plane.

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Linear Functions

QuadraticFunctions

Rational Functions

Inverse Functions




Example

The graph of f (x) ={

2x + 8 if x < −32 if x ≥ −3

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Linear Functions

QuadraticFunctions

Rational Functions

Inverse Functions




Given the graph of a function f , we can determine thedomain and range of f .Collect all the vertical lines that will intersect the graph.

The intersection of the region and the x-axis shows that thedomain of the function is the set of real numbers −2 ≤ x < 2since the point (2,1) is not a point in the graph anymore.

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Linear Functions

QuadraticFunctions

Rational Functions

Inverse Functions




To determine the range of the function, we consider theregion in the given figure which is the collection of all thehorizontal lines intersecting the graph. The intersection ofthe region and the y -axis gives the range of the function,−3 ≤ y < 1 .

BUSCALC

YvetteFajardo-Lim


Linear Functions

QuadraticFunctions

Rational Functions

Inverse Functions



Outline


BUSCALC

YvetteFajardo-Lim


Linear Functions

QuadraticFunctions

Rational Functions

Inverse Functions



Linear Functions

Definition

The slope of a line is defined as m =y2 − y1

x2 − x1, where

(x1, y1) and (x2, y2) are any two points on the line andx1 6= x2.

BUSCALC

YvetteFajardo-Lim


Linear Functions

QuadraticFunctions

Rational Functions

Inverse Functions



Linear Functions

Example

The slope of the line passing through the points (−1,3) and

(2,−2) is −53.

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YvetteFajardo-Lim


Linear Functions

QuadraticFunctions

Rational Functions

Inverse Functions



Linear Functions

DefinitionLinear functions are functions that have the formy = mx + b where m is the slope of the line and b is they-intercept. In the special case m = 0 with f (x) = b, we callthese functions constant functions with a horizontal line asits graph.

BUSCALC

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Linear Functions

QuadraticFunctions

Rational Functions

Inverse Functions



Slope-Intercept Form

DefinitionA linear equation in the form y = mx + b is called theslope-intercept form.

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Linear Functions

QuadraticFunctions

Rational Functions

Inverse Functions



Slope-Intercept Form

ExampleThe equation of the line with slope −3 and y-intercept 2 isgiven as y = −3x + 2.

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Linear Functions

QuadraticFunctions

Rational Functions

Inverse Functions



Point-Slope Form

If we know that a line passes through the point (x1, y1) andhas a slope of m then the point-slope form of the equationof the line is y − y1 = m(x − x1) .

Example

The equation of the line that passes through the point (2,3)

with slope32

is y =32

x.

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YvetteFajardo-Lim


Linear Functions

QuadraticFunctions

Rational Functions

Inverse Functions



Two-Point Form

The line through the points (x1, y1) and (x2, y2) is given by

the two-point form y − y1 =y2 − y1

x2 − x1(x − x1)

Example

The equation of the line passing through the points (−1,3)

and (2,−2) is y =− 5x + 4

3

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YvetteFajardo-Lim


Linear Functions

QuadraticFunctions

Rational Functions

Inverse Functions



General Form

Equations of the line can be reduced to the first degreeequation in the variables x and y of the formax + by + c = 0 where a,b, and c are real numbers and aand b are not both zero. This is called the general form ofthe equation of a line.

ExampleThe equation y = −3x + 2, can be written in the generalform 3x + y − 2 = 0.

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Linear Functions

QuadraticFunctions

Rational Functions

Inverse Functions



Intercept Form

The intercept form of the equation of a line is given byxa+

yb= 1.

ExampleThe equation 8x − 3y + 24 = 0 in its intercept form is

written asx3+

y8= 1.

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YvetteFajardo-Lim


Linear Functions

QuadraticFunctions

Rational Functions

Inverse Functions



Parallel and Perpendicular Lines

Equations of lines can also be determined given theequation of a parallel line or a perpendicular line. Two linesare parallel if they have the same slope and two lines areperpendicular when the product of their slopes is −1, thatis, their slopes are negative reciprocals.

BUSCALC

YvetteFajardo-Lim


Linear Functions

QuadraticFunctions

Rational Functions

Inverse Functions




Example

The equation of the line that passes through (−2,3) and isparallel to 4x − 3y = 2 is 4x − 3y + 17 = 0

BUSCALC

YvetteFajardo-Lim


Linear Functions

QuadraticFunctions

Rational Functions

Inverse Functions




Example

The equation of the line that passes through (1,−4) and is

perpendicular to y = −x2+ 4 is 2x − y − 6 = 0

BUSCALC

YvetteFajardo-Lim


Linear Functions

QuadraticFunctions

Rational Functions

Inverse Functions



Outline


BUSCALC

YvetteFajardo-Lim


Linear Functions

QuadraticFunctions

Rational Functions

Inverse Functions



Quadratic Functions

Definition

Functions of the form f (x) = ax2 + bx + c where a,b, c arereal numbers with a 6= 0 are called quadratic functions.The graph of a quadratic function is a parabola.

BUSCALC

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Linear Functions

QuadraticFunctions

Rational Functions

Inverse Functions



Quadratic Functions

Below is the graph of y = x2

BUSCALC

YvetteFajardo-Lim


Linear Functions

QuadraticFunctions

Rational Functions

Inverse Functions



Quadratic Functions

The graph of y = x2 shows that the function has a lowestpoint at (0,0). This lowest point is called the vertex. Thevertex is the lowest point if a > 0 and the parabola opensupward; if a < 0, the vertex is the highest point and theparabola opens downward.

The vertex is at point

(−

b2a

, f

(−

b2a

))while the axis of

symmetry is the line x = −b2a

.

BUSCALC

YvetteFajardo-Lim


Linear Functions

QuadraticFunctions

Rational Functions

Inverse Functions



Quadratic Functions

The following table serves as an aid in sketching the graphof the quadratic function f (x) = ax2 + bx + c.a > 0 parabola opens upwarda < 0 parabola opens downward

vertex

(−

b2a

, f

(−

b2a

))b2 − 4ac > 0 parabola has two x-interceptsb2 − 4ac = 0 parabola has one x-interceptb2 − 4ac < 0 parabola has no x-interceptx-intercepts solutions of 0 = ax2 + bx + cy -intercept c

BUSCALC

YvetteFajardo-Lim


Linear Functions

QuadraticFunctions

Rational Functions

Inverse Functions



Quadratic Functions

Example

Given the function f (x) = 2x2 − x − 2.a = 2 > 0 parabola opens upward

vertex

(14,−

178

)b2 − 4ac = 17 > 0 parabola has two x-intercepts

x-intercepts1±√

174

y-intercept −2

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YvetteFajardo-Lim


Linear Functions

QuadraticFunctions

Rational Functions

Inverse Functions



Quadratic Functions

Graph of the function f (x) = 2x2 − x − 2.

BUSCALC

YvetteFajardo-Lim


Linear Functions

QuadraticFunctions

Rational Functions

Inverse Functions



Standard Form of Quadratic Functions

Quadratic functions may also be expressed in the formf (x) = a(x − h)2 + k ,a 6= 0. This is referred to as standardform with vertex at the point (h, k).

Example

The quadratic function f (x) = 2x2 − x − 2 in standard form

is written f (x) = 2

(x −

14

)2

−178

.

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YvetteFajardo-Lim


Linear Functions

QuadraticFunctions

Rational Functions

Inverse Functions



Domain and Range of Quadratic Functions

The domain of a quadratic function is the set of realnumbers. If the parabola opens upward the range of thefunction is y ≥ k and if it is downward, the range is y ≤ k .

Example

The domain of the quadratic function f (x) = 2x2 − x − 2 is

the set of real numbers while its range is y ≥ −178

.

BUSCALC

YvetteFajardo-Lim


Linear Functions

QuadraticFunctions

Rational Functions

Inverse Functions



Vertical Shift

Given the function f (x), the graph of y = f (x) + k can beobtained by shifting the graph of f (x), k units up if k > 0while if k < 0, the graph is shifted k units down.

BUSCALC

YvetteFajardo-Lim


Linear Functions

QuadraticFunctions

Rational Functions

Inverse Functions



Horizontal Shift

Consider f (x − h) when y = f (x) . If h > 0, the graphundergoes a horizontal shift h units to the right; if h < 0, thegraph undergoes a horizontal shift h units to the left.

BUSCALC

YvetteFajardo-Lim


Linear Functions

QuadraticFunctions

Rational Functions

Inverse Functions



Outline


BUSCALC

YvetteFajardo-Lim


Linear Functions

QuadraticFunctions

Rational Functions

Inverse Functions



Rational Functions

DefinitionBy a rational function f (x) we mean a function whose

assignment rule is of the form f (x) =p(x)q(x)

, where p(x) and

q(x) are polynomials and q(x) 6= 0 .

The domain of the rational functions is the set of realnumbers except for the values which will make q(x) = 0.

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Linear Functions

QuadraticFunctions

Rational Functions

Inverse Functions



Rational Functions

To graph rational functions the asymptotes must bedetermined. The vertical asymptotes occur at the domainrestriction. If p(x) = anxn + . . .+ a2x2 + a1x + a0 andq(x) = bmxm + . . .+ b2x2 + b1x + b0, the following table is asummary to aid in sketching the graph.q(r) = 0, r ∈ R vertical asymptote is the line x = rn < m horizontal asymptote is the line y = 0

n = m horizontal asymptote is the line y =an

bmn > m no horizontal asymptote

n = m + 1 oblique asymptote is the line y =p(x)q(x)

BUSCALC

YvetteFajardo-Lim


Linear Functions

QuadraticFunctions

Rational Functions

Inverse Functions



Rational Functions

Example

Given the function f (x) =2x + 1x − 3

. The domain is the set of

all real numbers such that x 6= 3 . Thus, the verticalasymptote is x = 3 and since both the numerator anddenominator are linear, the horizontal asymptote is the liney = 2 .

BUSCALC

YvetteFajardo-Lim


Linear Functions

QuadraticFunctions

Rational Functions

Inverse Functions



Rational Functions

f (x) =2x + 1x − 3

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Linear Functions

QuadraticFunctions

Rational Functions

Inverse Functions



Rational Functions

f (x) =x2 − 4x − 3

BUSCALC

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Linear Functions

QuadraticFunctions

Rational Functions

Inverse Functions



Outline


BUSCALC

YvetteFajardo-Lim


Linear Functions

QuadraticFunctions

Rational Functions

Inverse Functions



Inverse Functions

DefinitionA function f is said to be one-to-one if every number in itsrange corresponds to exactly one number in its domain, thatis for all x1 and x2 in the domain of f , if x1 6= x2 thenf (x1) 6= f (x2) . Equivalently, f (x1) = f (x2) only when x1 = x2.

Showing that a function is one-to-one is often a tedious andoften difficult. However, if the value of f (x) increases as thevalue of x increases for all x in its domain, then the functionis one-to-one. Similarly, if the value of f (x) decreases as thevalue of x increases for all x in its domain, the function isone-to-one.

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Linear Functions

QuadraticFunctions

Rational Functions

Inverse Functions



Inverse Functions

Definition

If f is a one-to-one function then there is function f−1 , calledthe inverse of f , where f−1(f (x)) = x and f (f−1(x)) = x forall values of x in their respective domains. The domain off−1 is the range of f and the range of f−1 is the domain of f .

BUSCALC

YvetteFajardo-Lim


Linear Functions

QuadraticFunctions

Rational Functions

Inverse Functions



Inverse Functions

Example

The function f (x) = 3x − 5 is a linear function with a positiveslope and is an increasing function. Hence, its inverse

exists and f−1(x) =x + 5

3.

BUSCALC

YvetteFajardo-Lim


Linear Functions

QuadraticFunctions

Rational Functions

Inverse Functions



Inverse Functions

In general, the inverse of quadratic functions does not exist.However, if the domain will be restricted in such a way thatthe function is one-to-one on the restricted interval of thedomain, the function will have its inverse.

Example

The inverse of the function f (x) = x2 − 5, x ≤ 0 isf−1(x) = −

√x + 5.

BUSCALC

YvetteFajardo-Lim


Linear Functions

QuadraticFunctions

Rational Functions

Inverse Functions



Graph of Inverse Functions

The graph of an inverse is the reflection of the original graphabout the line y = x .

BUSCALC

YvetteFajardo-Lim


Linear Functions

QuadraticFunctions

Rational Functions

Inverse Functions



Outline


BUSCALC

YvetteFajardo-Lim


Linear Functions

QuadraticFunctions

Rational Functions

Inverse Functions



Exponential Functions

DefinitionThe exponential function with base a is defined for all realnumbers x by f (x) = ax , where a > 0, and a 6= 1 .

BUSCALC

YvetteFajardo-Lim


Linear Functions

QuadraticFunctions

Rational Functions

Inverse Functions




Example

The function f (x) = 2x is an exponential function with thegiven graph below.

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Linear Functions

QuadraticFunctions

Rational Functions

Inverse Functions




The graph of the function f (x) =

(12

)x

is shown below.

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Linear Functions

QuadraticFunctions

Rational Functions

Inverse Functions



Properties of Exponential Functions

1 The domain of f (x) = ax is the set of real numbers.2 The range of f (x) = ax is the set of positive real

numbers.3 The y -intercept of the graph of f (x) = ax is at point

(0,1), that is f (0) = 1.4 If 0 < a < 1, as the value of x increases, the value of y

decreases.5 If a > 1, as the value of x increases, the value of y

increases.6 f (x) is one-to-one, if ax = ay then x = y .7 The graph of f (x) has a horizontal asymptote y = 0.

BUSCALC

YvetteFajardo-Lim


Linear Functions

QuadraticFunctions

Rational Functions

Inverse Functions



Natural Exponential Function

The function f (x) = ex is called the natural exponentialfunction, where e ≈ 2.718281828. The graph of f (x) = ex

and f (x) = e−x is shown below.

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YvetteFajardo-Lim


Linear Functions

QuadraticFunctions

Rational Functions

Inverse Functions



Outline


BUSCALC

YvetteFajardo-Lim


Linear Functions

QuadraticFunctions

Rational Functions

Inverse Functions



Logarithmic Functions

DefinitionLet a > 0 and a 6= 1. The logarithmic function with base aand written as loga , is defined by y = loga x if and only ifx = ay for every x > 0 and every real number y.

Example

log5 25 = 2 and log4164

= −3

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Linear Functions

QuadraticFunctions

Rational Functions

Inverse Functions




The logarithmic function with base 10 is called the commonlogarithm and we commonly write this as log x , that is, ifthe base is omitted, it is understood to be 10. Anotherspecial logarithmic function is the natural logarithm withbase e written as ln x .

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Linear Functions

QuadraticFunctions

Rational Functions

Inverse Functions




The functions ax and loga x are inverse functions. Hence,the graph of loga x is simply reflecting the graph of ax aboutthe line y = x .

The graph below shows 2x and log2 x

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Linear Functions

QuadraticFunctions

Rational Functions

Inverse Functions



Properties of Logarithmic Functions

1 The domain of f (x) = loga x is the set of positive realnumbers.

2 The range of f (x) = loga x is the set of real numbers.3 The x-intercept of the graph of f (x) = loga x is at point

(1,0), that is f (1) = 0.4 If 0 < a < 1, as the value of x increases, the value of y

decreases.5 If a > 1, as the value of x increases, the value of y

increases.6 f (x) is one-to-one, if loga x = loga y then x = y .7 The graph of f (x) has a vertical asymptote x = 0.

Calculus by Beamer

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