m53 lec1.0 review of functions

Review of Functions

Mathematics 53

Institute of Mathematics (UP Diliman)

Institute of Mathematics (UP Diliman) Review of Functions Mathematics 53 1 / 43

For today

1 Functions

2 Basic Types of Functions

3 Constructing a Table of Signs

4 Operations on Functions

5 Piecewise-defined Functions

6 Functions as Mathematical Models


Functions

DefinitionLet X and Y be nonempty sets. A function f from X to Y, denoted f : X Y, isa rule that assigns to each element of X a unique element of Y.

X: domain of f , denoted dom fY: codomain of fThe set of all elements of Y that are assigned to some element of X is therange of f , denoted ran f .


Real-valued functions of a single variable

Real-valued functions of a single variable:Codomain: R

Math 53 deals with functions whose domain and range are subsets of R.

If the domain is not explicitly specified:Domain: dom f = {x R | f (x) is a real number}

Examples:1 f (x) = x2 dom f = R

2 f (x) =x2 2x 3

x + 1dom f = R \ {1}


Basic types of functions

Polynomial Functions - functions of the form

f (x) = anxn + an1xn1 + + a1x + a0

where n W, an, an1, ..., a0 are real numbers, with an 6= 0.

leading coefficient: andegree of f : ndom f = R polynomial functionsGraphs of polynomial functions: Unit 3


Some Examples of Polynomial Functions

1. Constant Functions - functions of the form f (x) = c, where c is a real numberdom f = R; ran f = {c}graph: horizontal line intersecting the y-axis at y = c

2. Linear Functions - functions of the form

f (x) = mx + b

with m 6= 0dom f = R; ran f = Rgraph: m is slope; y-intercept is bIf m > 0: line slants to the rightIf m < 0: line slants to the left


Some Examples of Polynomial Functions

3. Quadratic Functions - functions of the form

f (x) = ax2 + bx + c

with a 6= 0dom f = R

graph: parabola with vertex at( b2a ,

4acb24a

)If a > 0: parabola opens upward, ran f =

[4acb2

4a ,+)

If a < 0: parabola opens downward, ran f =(, 4acb24a

]Extreme function values of a quadratic function:

a > 0: f has a minimum function valuea < 0: f has a maximum function value

The extreme function value of f occurs at x = b2a and the extreme functionvalue of f is f

( b2a

)= 4acb

2

4a



Rational Functions - functions of the form

f (x) =p(x)q(x)

,

where p and q are polynomial functions, and q is not the constant zero function.

Domain: {x R | q(x) 6= 0}Graphs of general rational functions: Unit 3


Graphs of Functions

Example

The graph of f (x) =x2 2x 3

x + 1:

f (x) =x2 2x 3

x + 1=

(x 3)(x + 1)(x + 1)

= x 3 if x 6= 1

Domain: R \ {1}Range: R \ {4}Zero: x = 3Positive: (3,+)Negative: (,1) (1, 3)

3 2 1 1 2 3 4

5

4

3

2

1

1

0



Functions involving rational exponents or radicals - functions of the form

f (x) = n

x = x1/n

n is odd: dom f = Rn is even: dom f = [0, )



ExampleSquare root function: f (x) =

x

y =

x = y2 = x, y 0

1 2 3 4

2

1

1

2

0

The graph of x = y2

1 2 3 4

2

1

1

2

0

The graph of y =

x

The graph of f (x) =

x is the upper branch of the parabola x = y2


Other Functions involving Radicals

Example

Sketch the graph of g(x) =

2 x.

y =

2 xy2 = 2 x, y 0x = 2 y2, y 0

The graph of g is the lower branch of the parabola x = 2 y2.



Example

Sketch the graph of g(x) =

2 x.

2 1 1 2

2

1

1

2

0



Example

Graph h(x) =

4 x2.

The graph of h is the upper half of the graph of x2 + y2 = 4.

2 1 1 2

2

1

1

2



Trigonometric/Circular Functions

sine, cosine, tangent, cotangent, secant and cosecant functions

In Math 53, the trigonometric functions are viewed as functions on the set ofreal numbers.


Constructing a table of signs

A table of signs shows when a given mathematical expression is positive, zero ornegative.

Two Methods:

1 Interval Method

2 Test Value Method

In both cases, one must determine the numbers where the given mathematicalexpression is zero or undefined.



Example

Determine the intervals where the the graph of f (x) =2x2 x3

2x2 3x + 1 lies abovethe x-axis.

We want to determine the intervals for which

2x2 x32x2 3x + 1 > 0


Constructing a table of signsInterval Method: rewrite the expression as a product of factors whose table ofsigns are easily determined.

2x2 x32x2 3x + 1 =

x2(2 x)(2x 1)(x 1)

Zero at: x = 0, 2, Undefined at: x = 12 , 1

(, 0)(

0, 12) (

12 , 1)

(1, 2) (2,+)

x2 + + + + +2x 1 + + +x 1 + +2 x + + + +

x2(2 x)(2x 1)(x 1) + + +

The graph of f lies above the x-axis in the intervals (, 0) (

0, 12) (1, 2).



Test Value Method: test a value in the specified interval

(, 0)(

0, 12) (

12 , 1)

(1, 2) (2,+)

x2(2 x)(2x 1)(x 1) + + +

Sample point in (, 0): x = 1 (1)2(3)

(3)(2)

Sample point in(

0, 12)

:(+)(+)

()()We get the same result:

The graph of f lies above the x-axis in the intervals (, 0) (

0, 12) (1, 2).



Example

Find the domain of f (x) =5x

x2 1 .

Domain: x R such that 5xx2 1 0

Zero at: x = 0, Undefined at: x = 1, 1

(,1) (1, 0) (0, 1) (1,+)5x

x2 1 + +

Therefore,dom f = (,1) [0, 1)


Operations on Functions

Definition (Operations on Functions)Let f and g be functions, c R.

1 Addition: ( f + g)(x) = f (x) + g(x); dom( f + g) = dom f dom g2 Subtraction: ( f g)(x) = f (x) g(x); dom( f g) = dom f dom g3 Multiplication: ( f g)(x) = f (x)g(x); dom( f g) = dom f dom g

4 Division:(

fg

)(x) =

f (x)g(x)

;

dom(

fg

)= (dom f dom g) \ {x dom g | g(x) = 0}

5 Composition: ( f g)(x) = f (g(x));dom( f g) = {x dom g | g(x) dom f }

6 Scalar Multiplication: c f (x) = c ( f (x)); dom c f = dom f



Example

Express the function F(x) = sin2 (3x 1) as a composition of three functionslisted among the basic types of functions.

Let

f (x) = x2

g(x) = sin xh(x) = 3x 1

ThenF(x) = ( f g h) (x)



Example

Let f (x) = x2 and g(x) = x + h, where h is a nonzero constant. Find1h[( f g) (x) f (x)].

1h[( f g) (x) f (x)] = 1

h[ f (g(x)) f (x)]

=f (x + h) f (x)

h

=(x + h)2 x2

h

=(x2 + 2xh + h2) x2

h

=2xh + h2

h= 2x + h


Piecewise-defined functions

Piecewise-defined functions are functions that are defined by more than oneexpression. Such functions can be written in the form

f (x) =

f1(x), if x X1f2(x), if x X2

... if...

fn(x), if x Xn

where X1, ..., Xn R with Xi Xj = for all i 6= j.


The Absolute Value Function

Absolute Value Function - denoted by |x| and defined by

|x| =

x2 ={

x, x 0x, x < 0



The graph of f (x) = |x|

3 2 1 1 2 3

1

2

3

0



ExampleWrite f (x) = |2x 3|+ x as a piecewise function.

Recall that |x| ={

x if x 0x if x < 0

.

f (x) = |2x 3|+ x =

(2x 3) + x, if 2x 3 0(2x 3) + x, if 2x 3 < 0=

3x 3, if x 32

x + 3, if x < 32


The Greatest Integer Function

Greatest Integer Function (GIF) [[x]]: greatest integer less than or equal to x

Example1 [[2.4]] = 22 [[2]] = 2

3 [[2.1]] = 34 [[]] = 4



As a piecewise function:

[[x]] =

......

1, 1 x < 00, 0 x < 11, 1 x < 22, 2 x < 3...

...

In general,

[[x]] = n, for n x < n + 1 where n Z



The graph of f (x) = [[x]]

4 3 2 1 1 2 3 4

4

3

2

1

1

2

3

4

0



ExampleWrite f (x) = [[3x]] as a piecewise function.

f (x) = [[3x]] = n for n 3x < n + 1n3 x < n + 1

3

f (x) = [[3x]] =

......

0, 0 x < 131, 13 x

m53 lec1.0 review of functions

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