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Topic: Is (t^2-9)/(t-3) defined at t=3?
Replies: 166   Last Post: Oct 30, 2013 9:41 AM

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quasi

Posts: 10,451
Registered: 7/15/05
Re: Is (t^2-9)/(t-3) defined at t=3?
Posted: Sep 29, 2013 1:46 AM
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Peter Percival wrote:
>quasi wrote:
>> Hetware wrote:
>>> quasi wrote:
>>>> Hetware wrote:
>>>>>
>>>>> I'm reading a 1953 edition of Thomas's Calculus and
>>>>> Analytic Geometry.
>>>>>
>>>>> In it he states that given:
>>>>>
>>>>> F(t) = (t^2-9)/(t-3)
>>>>>
>>>>> F(t) = (t-3)(t+3)/(t-3) = t+3 when t!=3.
>>>>>
>>>>> But F(t) is not defined at t=3 because it evaluates to 0/0.
>>>>>
>>>>> If someone were to ask me if (t^2-9)/(t-3) is defined when t=3,
>>>>> I would say it is

>>>>
>>>> Then you would be wrong.
>>>>

>>>>> because it can be simplified to t+3.
>>>>
>>>> To get that result, you had to cancel the common factor t-3 in
>>>> numerator and denominator. But that cancellation depends on the
>>>> simplification
>>>>
>>>> (t - 3)/(t - 3) = 1
>>>>
>>>> which is only valid if t != 3.
>>>>
>>>> In a first level algebra course (Elementary Algebra) where
>>>> function concepts are not yet in play, the simplification
>>>>
>>>> (t^2 - 9)/(t - 3)
>>>>
>>>> = (t + 3)(t - 3))/(t - 3)
>>>>
>>>> = t + 3
>>>>
>>>> is allowed, without worrying about exceptional values of t for
>>>> which the simplification fails.
>>>>
>>>> But at the next level of algebra, algebraic expressions are
>>>> often being regarded as functions, so more care is taken to
>>>> identify those exceptional values.
>>>>

>>>>> Am I (and/or Thomas) engaging in meaningless hair-splitting
>>>>> regarding the question of F(3) being defined?

>>>>
>>>> For functions, identifying the precise domain is key.
>>>>
>>>> Thomas is correct.
>>>>
>>>> Those hairs _need_ to be split.
>>>>
>>>> All modern precalculus and calculus texts are careful (in the
>>>> context of deciding whether functions are equal) to identify
>>>> exceptional values where simplifications fail.

>>>
>>> The reason I have some misgiving about saying that F(3) is
>>> undefined is

>>
>> F is being regarded as a _function_

>
>With what domain and range?


The domain of F is all real numbers except 3.

The implied codomain of F is all real numbers.

The range of F is all real numbers except 6.

In Precalculus and Calculus level texts, it's understood that
the domain of a function of one variable is all of R unless
there are explicitly stated and/or implied domain restrictions.
In this case, the implied domain restriction is t != 3.

quasi


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9/28/13
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Hetware
9/28/13
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Michael F. Stemper
9/28/13
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quasi
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Hetware
9/28/13
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quasi
9/28/13
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Peter Percival
9/29/13
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quasi
9/28/13
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Richard Tobin
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9/29/13
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Hetware
10/6/13
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Hetware
10/6/13
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Peter Percival
10/6/13
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Hetware
10/6/13
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quasi
10/8/13
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quasi
10/7/13
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Peter Percival
9/29/13
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Michael F. Stemper
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Hetware
9/29/13
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quasi
9/29/13
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Hetware
9/29/13
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magidin@math.berkeley.edu
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Hetware
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magidin@math.berkeley.edu
10/7/13
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Hetware
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Peter Percival
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10/12/13
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fom
10/13/13
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Richard Tobin
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fom
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quasi
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quasi
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quasi
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fom
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fom
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quasi
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Peter Percival
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Hetware
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quasi
10/16/13
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10/16/13
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quasi
10/19/13
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Hetware
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quasi
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Hetware
10/20/13
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Virgil
10/18/13
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10/19/13
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