Post on 15-Jul-2015
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
The Modeling Process
Real-worldProblems
MathematicalModel
MathematicalConclusions
Real-worldPredictions
modelsolve
interpret
test
Shadows Forms
Outline
Modeling
Classes of FunctionsLinear functionsQuadratic functionsCubic functionsOther power functionsRational functionsTrigonometric FunctionsExponential and Logarithmic functions
Transformations of Functions
Compositions of Functions
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
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
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
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.
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.
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
Classes of FunctionsLinear functionsQuadratic functionsCubic functionsOther power functionsRational functionsTrigonometric FunctionsExponential and Logarithmic functions
Transformations of Functions
Compositions of Functions
Transformations of Functions
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.
Transformations of Functions
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
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 unitsupward
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
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 unitsupward
I y = f (x)− c , shift the graph of y = f (x) a distance c unitsdownward
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
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 unitsupward
I y = f (x)− c , shift the graph of y = f (x) a distance c unitsdownward
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 units
to the left
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 unitsupward
I y = f (x)− c , shift the graph of y = f (x) a distance c unitsdownward
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
Classes of FunctionsLinear functionsQuadratic functionsCubic functionsOther power functionsRational functionsTrigonometric FunctionsExponential and Logarithmic functions
Transformations of Functions
Compositions of Functions
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.