Fcn Algebra - Intermediate Algebra - Lecture Slides, Slides of Algebra

Download Fcn Algebra - Intermediate Algebra - Lecture Slides and more Slides Algebra in PDF only on Docsity!

§2.3 Algebra

of Functions

Review §

 Any QUESTIONS About

• §2.2 → Function Graphs

 Any QUESTIONS About HomeWork

• §2.2 → HW-

MTH 55

Function Analogy → Machinery

 The function pictured has been named f. Here x represents an arbitrary input, and f ( x ) (read “ f of x ,” “ f at x ,” or “the value of “ f at x ”) represents the corresponding output.

Implicit Domain

• If the domain of a function that is

defined by an equation is not explicitly

specified, then we take the domain of

the function to be the LARGEST SET OF

REAL NUMBERS that result in REAL

NUMBERS AS OUTPUTS.

• i.e., DEFAULT Domain is all x ’s that produce VALID Functional RESULTS

Example  Find Function Domain

• This Fcn Undefined for x − 8 = 0

f x

x

 To determine what x-value would cause x − 8 to be 0, we solve the equation:

 Thus 8 is not in the domain of f , whereas all other real numbers are.  Then the domain of f is

8

8 0 ⇒ =

− = x

x

{ x | x is a Real No. and x ≠ 8 }

Algebra of Functions

• The Sum, Difference, Product, or Quotient of

Two Functions

• Suppose that a is in the domain of two

functions, f and g. The input a is paired with

f ( a ) by f and with g ( a ) by g.

• The outputs can then be added to obtain: f ( a )

+ g ( a ).

Example  Function Algebra

• Find the Following for these Functions

f ( ) x = 2 x − x^2 and g x ( ) = 3 x +1,

• a) ( f + g )(4) b) ( f − g )( x )
• c) ( f / g )( x ) d) ( f • g )(−1)

 SOLUTION

a) Since f (4) = −8 and g (4) = 13, we have ( f + g )(4) = f (4) + g (4) = −8 + 13 = 5.

Example  Function Algebra

c) ( f / g )( x ) →

( f − g )( ) x = f ( ) x − g x ( )

= 2 x − x^2 − (3 x +1)

= − x^2 − x −1.

( f / g )( ) x = f ( ) / x g x ( )

x x

x

1 3

x ≠ −

Assumes

f ( ) x = 2 x − x^2 and g x ( ) = 3 x +1, b) ( f − g )( x ) →

 SOLUTION for

Example  Function Algebra

• Given f ( x ) = x^2 + 2 and g ( x ) = x − 3,

find each of the following.

• a) The domain of f + g , f − g , f • g , and f / g
• b) ( f − g )( x ) c) ( f / g )(x)
• SOLUTION a)
• The domain of f is the set of all real numbers. The domain of g is also the set of all real numbers. The domains of f + g , f − g , and f • g are the set of numbers in the intersection of the domains; i.e., the set of numbers in both domains, or all real No.s
• For f / g , we must exclude 3, since g (3) = 0

Example  Function Algebra

• SOLUTION b) → ( f − g )( x )
  • ( f − g )( x ) = f ( x ) − g ( x ) = ( x^2 + 2) − ( x − 3) = x^2 − x + 5
• SOLUTION c) → ( f / g )( x )

2

f x

f g x

g x

x

x

• Remember to add the restriction that x ≠ 3, since 3 is not in the domain of ( f / g )( x )

P2.3-56 Functions by Graphs

**-

-**

0

1

2

3

4

5

6

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

M55_§2.2_Graphs_0806.xls

y = g ( ) x

x

y

y = f^ ( ) x x^ f(x)^ g(x) -5 - -4 5 0 -3 4 1 -2 3 2 -1 3 2 0 2 0 1 1 1 2 -1 1 3 - 4 - 5 -

 From the Graph the fcn T-table

P2.3-56 Graph ( f − g )( x )

• Recall from Lecture - ( f − g )( x ) = f(x ) − g( x )
• Next Use for Plotting - y 1 = f(x ) - y 2 = g(x )
• Thus for Plot
  • f(x ) − g( x ) = y 1 − y 2 - - - - - -

0

1

2

3

4

5

6

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

M55_§2.2_Graphs_0806.xls

y (^) 2 = g ( ) x x

y = f ( ) x^ y 1

**-

-**

0

1

2

3

4

5

6

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

M55_§2.2_Graphs_0806.xls

x

y^ P2.3-56 (

f

g

x

) Graph

y = g^ ( ) x

y = f ( ) x

y =^ (^ f − g )( ) x

All Done for Today

Fcn Algebra

By

MicroProcessor