1.2 the Algebra of Functions

7/28/2019 1.2 the Algebra of Functions

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The Algebra of Functions andGraphs

Definition of a Function

Domain and Range

Function Notation


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Definition of Function

A function is a relation that has at most onevalue ofy for each unique value ofx.

{(1, 2), (3, 4), (5, 6), (7, 8)} is a functionsince there is only one value of y for eachdifferent value of x

{(1, 2), (3, 3), (1, 4), (5, 6)} is not a function

since for x= 1, there are two possible valuesfor y, 2 or 4.


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Example #1

From the graph we see that the line extends

to the left to negative infinity and to the rightto positive infinity.

The domain is all real numbers .

The range is all real numbers .

In interval notation

domain (-,)

range is (-,)

What is the range and domain of y = x +2?


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Example #2

The set of all x values for this equation is allReal numbers.

The set of all possible y values is all positivenumbers from zero to positive infinity.

Domain (-, )

Range [0,)

What is the domain ofy= x2?


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Domain of a Complex Function

When dealing with complex functions wewrite the domain as an exclusion.

In other words what x values are not in thedomain.

By setting the denominator equal to zero andsolving for x we find the x values that are notpart of the domain.

These x values are also the asymptotes of theequation.


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Example #3

What is the domain of1( )

1f x

x

Set the denominator NOT equal to zero

and solvex+10 x -1

The domain is {x|x

-1}(Notice the asymptote at x=-1)


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Example #4

What is the domain of ( ) 2f x x

Set radicand greater than or equal to zero

and solvex+2>0 x > -2

The domain is {x|x>-2}


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Function Notation

Writing an equation in function notation:

y= 4x - x2 f(x)= 4x - x2

Read as f of x is 4x - x2

You can use other letters to denote

functions g(x), t(x), v(t), etc.


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Writing in Function Notation

An example:

Find the domain:

2 2

1 1

4 5 4 5

x xy f x

x x x x

2 4 5 0

5 1 0

5,1

| 5,1

x x

x x

x

x x

Domain:


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Evaluating Functions

We evaluate a function the same waywe evaluate an equation substitution.

Given f(x) = x2 + x 5Find f(2) = x2 + x 5

f(2) =(2)2 + (2) 5

f(2) = 4 + 2 - 5 = 1


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How to Determine a FunctionBy comparing the domain with range or bygraphing

Is f(x) = -2x+2 a function?

Yes, this is a function. Only one value of y foreach unique x.

Graph this on the Y= screen

x

y

-6 -4 -2 0 2 4 6

-4

-2

0

2

4


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How to Determine a Function

How about an absolute value function

y=|x|

x

y

-6 -4 -2 0 2 4 6

-4

-2

0

2

4

Yes, it is a function. For each xonly one y


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Vertical Line Test

By using the Vertical Line Test we can

quickly determine if a relation is a function.

To be a function a vertical line cannotintersect its graph at more than one point.

Is this circle a function? y

x

NO!


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Vertical Line Test

Which Graphs are Functions?

y

x

y

x

x= y2 y = x2

No! Yes!

Algebraically: solve for y

y x


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Algebraic Determination

Is this circle a function?

y

x

NO!

2 2 1x y

2 2

2

1

1

y x

y x

1.2 the Algebra of Functions

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