Constant that makes g(x) continous over (-inf,inf)

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In summary, the task is to find the constant c that will make the function g(x) = x^2 - c^2 continuous over all real numbers. The function is composed of a quadratic and a polynomial, which are both continuous over all real numbers. The only possible point of discontinuity is at x = 4. To make the function continuous at this point, the limit of g(x) as x approaches 4 from the left must be equal to the limit of g(x) as x approaches 4 from the right. This leads to the equation c4 + 20 = 4^2 + c^2. Using the quadratic formula, c can be solved for and the function will be continuous over all real numbers
  • #1
JeffNYC
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Homework Statement



Find the C (constant) that makes the function g(x) continuous over all real numbers.


Homework Equations



g(x) = x^2 - c^2 | if x < 4

= cx + 20 | if x >= 4

The Attempt at a Solution



Since the function is composed of a quadratic and a polynomial they are continuous over all real numbers. Thus, the only possible point of discontinuity lies at x = 4.

lim g(x) as x -> 4 = 4c + 20

I need to make the limit of g(x) as x -> 4 from the left equal to the above limit.

So logically I need to solve for c

c4 + 20 = 4^2 + c^2

Assuming I've thought this through correctly up until this point...I guess I just can't solve the equation. Do I need to use the quadratic formula?

Thanks Guys,

Jeff
 
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  • #2
Quadratic formula sounds like a good place to start.
 
  • #3
Actually, I believe that thinking rather than the quadratic formula is a good place to start! Go back and check your equations. You should not have 4c+ 20= 42+ c! You've dropped a sign.
 
  • #4
Well, yeah, thinking is always good, but he *thought* he should try the quadratic formula and I think he's right in the sense that it will give him the correct answer.
 
  • #5
It would be better to think the correct equation first!

My real point was that if he had the right equation, it would be trivial to factor.
 

Related to Constant that makes g(x) continous over (-inf,inf)

1. What is a constant that can make a function continuous over the entire real number line?

The constant that can make a function continuous over the entire real number line is called the limit. This constant is crucial in ensuring that the function has a smooth and unbroken graph, without any gaps or jumps.

2. How does a constant affect the continuity of a function?

The constant, or limit, acts as a connecting point between different parts of the function. It ensures that the function remains smooth and unbroken, without any abrupt changes in its behavior.

3. Can any constant work as a limit to make a function continuous?

No, not all constants can work as a limit to make a function continuous. The constant must be carefully chosen to match the behavior of the function at the point of discontinuity.

4. How can I determine the appropriate constant to make a function continuous?

To determine the appropriate constant, you must first identify the point of discontinuity in the function. Then, you can use the limit definition to calculate the value of the constant that makes the function continuous at that point.

5. Is it possible to make a function continuous at all points using a constant?

In some cases, it may not be possible to make a function continuous at all points using a single constant. This is because the behavior of the function may be too complex and require different constants for different points of discontinuity. In such cases, the function may still be considered continuous over the entire real number line, but with multiple constants instead of just one.

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