Lesson 2: A Catalog of Essential Functions

Section 1.2A Catalog of Essential Functions

V63.0121, Calculus I

January 22, 2009

Announcements

I Blackboard is up

I First HW due Thursday 1/29

I ALEKS initial assessment due Friday 1/30

Outline

Modeling

Classes of FunctionsLinear functionsQuadratic functionsCubic functionsOther power functionsRational functionsTrigonometric FunctionsExponential and Logarithmic functions

Transformations of Functions

Compositions of Functions

The Modeling Process

Real-worldProblems

MathematicalModel

MathematicalConclusions

Real-worldPredictions

modelsolve

interpret

test

Plato’s Cave

The Modeling Process

Real-worldProblems

MathematicalModel

MathematicalConclusions

Real-worldPredictions

modelsolve

interpret

test

Shadows Forms

Outline

Modeling




Classes of Functions

I linear functions, defined by slope an intercept, point andpoint, or point and slope.

I quadratic functions, cubic functions, power functions,polynomials

I rational functions

I trigonometric functions

I exponential/logarithmic functions

Linear functions

Linear functions have a constant rate of growth and are of the form

f (x) = mx + b.

Example

In New York City taxis cost $2.50 to get in and $0.40 per 1/5 mile.Write the fare f (x) as a function of distance x traveled.

AnswerIf x is in miles and f (x) in dollars,

f (x) = 2.5 + 2x

Linear functions


f (x) = mx + b.

Example



f (x) = 2.5 + 2x

Linear functions


f (x) = mx + b.

Example



f (x) = 2.5 + 2x

Quadratic functions

These take the form

f (x) = ax2 + bx + c

The graph is a parabola which opens upward if a > 0, downward ifa < 0.

Quadratic functions

These take the form

f (x) = ax2 + bx + c

The graph is a parabola which opens upward if a > 0, downward ifa < 0.

Cubic functions

These take the form

f (x) = ax3 + bx2 + cx + d

Other power functions

I Whole number powers: f (x) = xn.

I negative powers are reciprocals: x−3 =1

x3.

I fractional powers are roots: x1/3 = 3√

x .

Rational functions

DefinitionA rational function is a quotient of polynomials.

Example

The function f (x) =x3(x + 3)

(x + 2)(x − 1)is rational.

Trigonometric Functions

I Sine and cosine

I Tangent and cotangent

I Secant and cosecant

Exponential and Logarithmic functions

I exponential functions (for example f (x) = 2x)

I logarithmic functions are their inverses (for examplef (x) = log2(x))

Outline

Modeling





Take the sine function and graph these transformations:

I sin(x +

π

2

)I sin

(x − π

2

)I sin (x) +

π

2

I sin (x)− π

2

Observe that if the fiddling occurs within the function, atransformation is applied on the x-axis. After the function, to they -axis.


Take the sine function and graph these transformations:

I sin(x +

π

2

)I sin

(x − π

2

)I sin (x) +

π

2

I sin (x)− π

2

Observe that if the fiddling occurs within the function, atransformation is applied on the x-axis. After the function, to they -axis.

Vertical and Horizontal Shifts

Suppose c > 0. To obtain the graph of

I y = f (x) + c , shift the graph of y = f (x) a distance c units

upward

I y = f (x)− c , shift the graph of y = f (x) a distance c units

downward

I y = f (x − c), shift the graph of y = f (x) a distance c units

to the right

I y = f (x + c), shift the graph of y = f (x) a distance c units

to the left



I y = f (x) + c , shift the graph of y = f (x) a distance c unitsupward

I y = f (x)− c , shift the graph of y = f (x) a distance c units

downward


to the right


to the left




I y = f (x)− c , shift the graph of y = f (x) a distance c unitsdownward


to the right


to the left





I y = f (x − c), shift the graph of y = f (x) a distance c unitsto the right


to the left





I y = f (x − c), shift the graph of y = f (x) a distance c unitsto the right

I y = f (x + c), shift the graph of y = f (x) a distance c unitsto the left

Outline

Modeling




Composition is a compounding of functions in succession

f g

g ◦ f

x (g ◦ f )(x)f (x)

Composing

Example

Let f (x) = x2 and g(x) = sin x . Compute f ◦ g and g ◦ f .

Solutionf ◦ g(x) = sin2 x while g ◦ f (x) = sin(x2). Note they are not thesame.

Composing

Example

Let f (x) = x2 and g(x) = sin x . Compute f ◦ g and g ◦ f .

Solutionf ◦ g(x) = sin2 x while g ◦ f (x) = sin(x2). Note they are not thesame.

Decomposing

Example

Express√

x2 − 4 as a composition of two functions. What is itsdomain?

SolutionWe can write the expression as f ◦ g, where f (u) =

√u and

g(x) = x2 − 4. The range of g needs to be within the domain off . To insure that x2 − 4 ≥ 0, we must have x ≤ −2 or x ≥ 2.

The Far Side

Lesson 2: A Catalog of Essential Functions

Education

formf x

answerif x

examplef x

fare f x

example f x

function of distance

examplethe function

sine function