Trig/Pre-Calculus

Opening Activity

532)(

2)(

33)(

2

xxxh

xxg

xxf

3)

2)

1)

Write the domain of the following functions.

Solve the following inequalities.

12362

953

082

xx

x

x

or 6)

5)

4)

The graph of a function f is the collection of ordered pairs (x, f(x)) where x is in the domain of f.

x

y

4

-4

(2, –2) is on the graph of f(x) = (x – 1)2 – 3.

(2, –2)

f(2) = (2 – 1)2 – 3 = 12 – 3 = – 2

x

y

4

-4

The domain of the function y = f (x) is the set of values of x for which a corresponding value of y exists.

The range of the function y = f (x) is the set of values of y which correspond to the values of x in the domain.

Domain

Range

x

y

– 1

1

Example: Find the domain and range of the function f (x) = from its graph.

The domain is [–3,∞).

The range is [0,∞).

3x

Range

Domain

(–3, 0)

• decreasing on an interval if, for any x1 and x2 in the interval, x1 < x2 implies f (x1) > f (x2),

• constant on an interval if, for any x1 and x2 in the interval, f (x1) = f (x2).

The graph of y = f (x):

• increases on (– ∞, –3),

• decreases on (–3, 3),

• increases on (3, ∞).

A function f is:• increasing on an interval if, for any x1 and x2 in the

interval, x1 < x2 implies f (x1) < f (x2),

(3, – 4)

x

y(–3, 6)

–2

2

A function value f(a) is called a relative minimum of f if there is an interval (x1, x2) that contains a such that

x1 < x < x2 implies f(a) f(x).

x

y

A function value f(a) is called a relative maximum of f if there is an interval (x1, x2) that contains a such that

x1 < x < x2 implies f(a) f(x).

Relative minimum

Relative maximum

Graphing Utility: Approximate the relative minimum of the function f(x) = 3x2 – 2x – 1.

– 6

– 6

6

6

– 0.86– 4.79

– 1.79

2.14

0.58 0.76

-3.24

-3.43

Zoom In:

Zoom In: The approximate minimum is

(0.67, –3.33).

Determine the relative minima and maxima of the following function. Determine where the graph is increasing, decreasing, and constant.

xxxxg 623

x

y

4

-4

A piecewise-defined function is composed of two or more functions.

f(x) =3 + x, x < 0 x2 + 1, x 0

Use when the value of x is less than 0.

Use when the value of x is greater or equal to 0.

(0 is not included.)open circle

(0 is included.)closed circle

A function f is even if for each x in the domain of f, f (– x) = f (x).

x

yf (x) = x2

f (– x) = (– x)2 = x2 = f (x)

f (x) = x2 is an even function.

Symmetric with respect to the y-axis.

A function f is odd if for each x in the domain of f, f (– x) = – f (x).

x

y

f (x) = x3

f (– x) = (– x)3 = –x3 = – f (x)

f (x) = x3 is an odd function.

Symmetric with respect to the origin.

Now we are going to graph the piecewise function from DNA #4-6 by HAND.

4,5

4,621

)(

xx

xxxf

x

y

4

-4

Vertical Line Test

A relation is a function if no vertical line intersects its graph in more than one point.

This graph does not pass the vertical line test. It is not a function.

This graph passes the vertical line test. It is a function.

y = x – 1x = | y – 2|

x

y

4

-4

Graph this…

2,5

2,32)(

xx

xxxf

1)

1,212,4

2,3)(

xxx

xxf

2)

112

2

xx

x

).()(&

).()(

).()(

2121

2121

2121

xfxfxxf

xfxfxxf

xfxfxxf

have we anyfor if interval an onconstant is function A

implies if interval an on decreasing is function A

implies if interval an on increasing is function A

Increasing, Decreasing, and Constant Functions

Consider…1)2()( 2 xxf 1)Ex 5)( xxf 2)Ex

Relative Minimum and Maximum Values.

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,)(

).()(,

)(

21

21

21

21

xfafxxxaxx

af

xfafxxxaxx

af

then ifthat such containing interval an exists there iffor

if maximum relative a is value function A

then ifthat such containing interval an exists there iffor

if minimum relative a is value function A

We will use a graphing utility to find the following functions relative minima and maxima.

23 2)( xxxf

EVEN Functions

)()(

:

xfxf

x

everyfor if EVEN is function A

Every EVEN function is symmetric about the y-axis.

ODD Functions

)()(

:

xfxf

x

everyfor if ODD is function A

Every ODD function is symmetric about the y-axis.

Now we are going to graph the piecewise function from DNA #4-6 by HAND.

4,5

4,621

)(

xx

xxxf

Graph this…

2,5

2,32)(

xx

xxxf

1)

1,212,4

2,3)(

xxx

xxf

2)

112

2

xx

x

)()()1(

)3(32 2

xfhxfxf

fxxxf

c. b. a.

for 6)

.0)4()4(

922

hh

fhf

xxxf

,

find ,For 5)

0132;

223 yxxxy of function a is if Determine 7)

13)(

x

xf :domain the Find 8)

Ex 4)The net sales for a car manufacturer were $14.61 billion in 2005 and $15.78 billion in 2006. Write a linear equation giving the net sales y in terms of x, where x is the number of years since 2000. Then use the equation to predict the net sales for 2007.

76.817.1

85.517.161.14517.161.14

17.1117.1

6578.1561.14

0

xy

xyxy

m

x

line the on pts 2 are 15.78 6, and 14.61 5, So,

2000 represents

2007for dollars billion 95.16$

95.1676.819.8

76.8717.1

yyy

Graph the following linear functions. Graph #1 – 3 on the same coordinate plane.

5)

4) 3)

2) 1)

3

16210

2010245

x

yyx

yxyx