chapter 1: section 1.1 functions and modelsmayaj/chapter1_sec1.1... · chapter 1: section 1.1...

Chapter 1: Section 1.1

Functions and Models

Section 1.1: Four Ways to Represent a Function

Definition: A Function is a rule that assigns to each input value x exactly out-

put value y = f(x). The variable x is called the variable, and y is called the

variable. The is the set of all allowable x values, and the

is the set of all possible y values.

An algebraic description of a function exists when there is an explicit formula for y = f(x).

Example 1: Find the domain, the range, the value of f(3) and sketch the graph of the following

functions:

(a) f(x) = x ...this is called the ”identity” function

(b) f(x) = 2x� 4

Example 2: Find the domain and range for each of the following

(a) f(x) =

p2x� 4 ...does f(1) exist here?

2×-4>-0

÷t¥⇒ :#

Chapter 1: Sec1.1, Four Ways to Represent a Function

(b) f(x) =

2xpx

2�4x...does f(1) exist here?

If an explicit formula does not exist then we can sometimes define a function graphically

Example 3, Reading the information from a graph: The graph of a function f is shown in the

following figure

(a) Find the values of f(�2), f(0) and f(2).

(b) What are the domain and range of f?

�3 �2 �1 0 1 2 3

�3

�2

�1

1

2

3

x

f(x)

Vertical Line Test: A curve in the xy-plane is the graph of a i↵ no vertical line

passes through the curve more than .

2 Spring 2017, Maya Johnson

XZ . 4× > 0

Yeats.

*XfIDomain:(-oo,o)U(4Tfl

1) does Not exist,

since × =1 not in domain

.

(a) fl - 2) =-1¥- µ

/ ]flz )=µ

g- ( b ) Domain : £2 , 3 ]

Raise :[ - I, 3 ]

functiononce


Example 3: Determine whether each is a function of x.

(a)

x

f(x)

(b)

x

f(x)

(c)

x

f(x)


/ || / / /Fx ' is a function

|||| / fsxsmisohnyfuohetinoieis

filled in, so only one y - value

for that x - value.

p.p.pe.

f↳ is Not a fund . ,


Example 4 Given the graph of f(x) and g(x) below, determine the following:

�4 �3 �2 �1 0 1 2 3 4

�4

�3

�2

�1

1

2

3

4

x

y

f(x)

g(x)

(a) f(1)

(b) g(0)

(c) The values(s) of x for which f(x) = �3

(d) The values(s) of x for which f(x) = g(x)

The four possible ways to represent a function are:

(1)

(2)

(3)

(4)

Example 5: Let f(x) = x

2+ 2x� 1 and g(x) =

1x�2 , find the following

(a) f(1)

(b) g(�8)

(c) f(x+ h)


,

.

IF' EHIII

Algebraically ( with a formula )

Through words ( in a word problem )

Graphically

Numerically ( a table of data )


(d)

f(x+h)�f(x)h

Definition: Functions whose definition involve more than one rule are called Piecewise Functions.

To graph, graph each rule over the appropriate portion of the domain.

Example 6, Absolute value function: The absolute value of a number x, denoted by |x|, is the

distance from x to 0 on the real number line. The plot and definition of absolute value function if given

below.

x

y

|x| =(x x � 0

�x x < 0

Example 7: Find the domain and sketch graph of the function

f(x) =

8<

:x+ 1 if x < 1

1� x if x � 1



Example 8: Find an expression for the function whose graph is the given curve

�4 �3 �2 �1 0 1 2 3 4

�1

1

2

3

4

x

f(x)

Symmetry:

If a function f satisfies f(�x) = f(x) for every number x in its domain, then f is called an

function. An even function is symmetric about the .

If f satisfies f(�x) = �f(x) for every number x in its domain, then f is called an function.

An odd function is symmetric about the .

Example 9: Determine whether each of the following functions is even, odd or neither even nor

odd.

(a) f(x) =

x

3

x

2+1

(b) f(x) = (x+ 1)|x|

(b) f(x) = x

4+ x

2+ 1


]04€

,-2 < xez

Zx - 3, 2 I x 14

r = 2.

Equation of a semi - Circle :

y -. ra ⇒y= FX

⇒ b = -

3|ftp.ntcoeo = I (2) tb

⇒ b = - 3


Example 10: Problem#60 Page 24 of the book

An electricity company charges its customers a base rate of 10 a month, plus 6 cents per kilowatt-

hour(kWh) for the first 1200 kWh and 7 cents per kWh for all usage over 1200 kWh. Express the

monthly cost E as a function of the amount x of electricity used. Then graph the function E for

0 x 2000.

Example 11: A closed rectangular box with a volume of 64 m

3has a square base.

(a) Express the surface area as a function of the length (x) of the sides of the base

(b) What is the domain?


.

:× ) = {

10 + .06 × ) o ex E 1200

82 +. 07 ( X - 1200 ) , X 71200

QQ HEBEI,ks¥tE÷;¥se±i¥÷#

X > 0 ⇒

Domadnisl¥


Increasing and Decreasing Functions:

A function f is called % on an interval I if f(x1) < f(x2) whenever x1 < x2 in

I. It is called & on an interval I if f(x1) > f(x2) whenever x1 < x2 in I.

We can see from the figure below that the function f(x) = x

2is decreasing on the interval

and increasing on the interval .

x

y


chapter 1: section 1.1 functions and modelsmayaj/chapter1_sec1.1... · chapter 1: section 1.1...

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