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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: Oct 14, 2013 3:58 PM
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As I've previously suggested, the wording is key ...

Consider the following problems ...

Problem (1)

Let f(x) = (x^2 - 9)/(x - 3).

(a) Is f(3) defined?

answer: No.

(b) Is f continuous on its domain?

answer: Yes.

(c) Is f continuous on the set of real numbers?

answer: No, f is not continuous at x = 3.

(d) Does lim (x->3) f(x) exist?

answer: Yes, the limit is 6.

(d) Does there exist a function g continuous on the set of
real numbers such that g(x) = f(x) for all x != 3?

answer: Yes, the function g(x) = x + 3 has that property,
and in fact, it's the only such function.

Problem (2):

If a function f(x) is continuous on the set of real numbers
and if f(x) = (x^2 - 9)/(x - 3) for all x != 3, must f(x)
be equal to x + 3 for all x?

answer: Yes.

Remarks:

In problem (2), the wording could be changed in the following
ways without affecting the meaning of the question.

Problem (2), variation (1):

Assume a function f(x) is continuous on the set of real numbers
and is such that f(x) = (x^2 - 9)/(x - 3) for all x != 3. Must
f(x) be equal to x + 3 for all x?

answer: Yes.

Problem (2), variation (2):

Suppose a function f(x) is continuous on the set of real numbers
and suppose f is such that f(x) = (x^2 - 9)/(x - 3) for all
x != 3. Must f(x) be equal to x + 3 for all x?

answer: Yes.

Problem (2), variation (3):

Given that a function f(x) is continuous on the set of real
numbers and f(x) = (x^2 - 9)/(x - 3) for all x != 3, must
f(x) be equal to x + 3 for all x?

answer: Yes.

However, the following rewording is not acceptable ...

Problem (2), variation (4):

Define a function f(x) to be continuous on the set of real
numbers and let f(x) = (x^2 - 9)/(x - 3) for all x != 3.
Must f(x) be equal to x + 3 for all x?

answer: The statement of the problem is flawed. You can't
_define_ a function to be continuous.

To fix the wording, replace "define" by "if", "assume",
"suppose", or "given", and the issue evaporates.

It's as simple as that.

quasi


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9/28/13
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