Trig/Pre-Calculus Opening Activity

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.

).()(

,)(

).()(,

)(

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