Are these two functions equivalent when y = 0?

In summary, "Equivalent" means that two functions have the same output when the input is 0. To determine equivalence at y = 0, you can plug in x = 0 and compare the resulting y-values. There are some special cases to consider, such as when one function is undefined at x = 0 while the other is defined. Two functions can be equivalent at y = 0 but not at other values of y, as equivalence at y = 0 only requires the same output at x = 0. There are other ways to compare two functions, but determining equivalence at y = 0 can provide useful information about their relationship.
  • #1
497
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Hello!

Am I right to conclude that:
[tex]y^2(A(\frac{x}{y})^2 + B(\frac{x}{y})+ C) = Ax^2 + Bxy + Cy^2[/tex]

Only when y does not equal zero. I'm guessing I could evaluate the function lim y -> 0 but this is not the same as y explicitly equalling zero, is it? On the RHS, y can equal zero, on the LHS, y cannot equal zero. So I am guessing they are only equal when y does not equal zero.
Is this true?

Many thanks.
 
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  • #2
Indeed, you cannot evaluate the expression in 0. It is undefined in 0.
 
  • #3
Thanks Micro!
 

1. What is the meaning of "equivalent" in this context?

"Equivalent" means that the two functions have the same output, or y-value, when the input, or x-value, is 0.

2. How can I determine if two functions are equivalent when y = 0?

You can determine if two functions are equivalent when y = 0 by plugging in x = 0 into both functions and comparing the resulting y-values. If they are the same, then the functions are equivalent.

3. Are there any special cases or exceptions to consider when determining equivalence at y = 0?

Yes, there are some special cases to consider. One example is when one function is undefined at x = 0, while the other is defined. In this case, the functions are not equivalent at y = 0.

4. Can two functions be equivalent at y = 0 but not at other values of y?

Yes, it is possible for two functions to be equivalent at y = 0 but not at other values of y. This is because equivalence at y = 0 only requires that the functions have the same output when x = 0, but they may differ at other values of x.

5. Is determining equivalence at y = 0 the only way to compare two functions?

No, there are other ways to compare two functions, such as graphing them and analyzing their behavior at different values of x. However, determining equivalence at y = 0 is one way to compare the functions at a specific point and can provide useful information about their relationship.

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