- #1
Nikitin
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- 27
Some little detail has been giving me a head-ache for 1 of my problems.. And here is why.
Does ((-1)^{2})^{1/2} equal 1 or -1? You decide...
Does ((-1)^{2})^{1/2} equal 1 or -1? You decide...
No. But if you asked the questionIsn't the better answer +-1??
Incorrect simplification is a pervasive problem. People make this mistake with the trigonometric functions and their "inverses" a lot too.Seriously, a problem like this thing ruined my entire evening.. I couldn't figure out what was wrong when I was simplifying a ridiculously long expression during my math homework...
The mathematical concept of a function expresses the intuitive idea that one quantity (the argument of the function, also known as the input) completely determines another quantity (the value, or the output). A function assigns exactly one value to each input of a specified type.
So one can have different functions that have the same input? Like, -1 and 1 being an output in my question depending on how you read it?
So one can have different functions that have the same input? Like, -1 and 1 being an output in my question depending on how you read it?
Some little detail has been giving me a head-ache for 1 of my problems.. And here is why.
Does ((-1)^{2})^{1/2} equal 1 or -1? You decide...
If you mean, "Can a function have two different outputs for the same input?", then no. It's part of the definintion of a function that it has only one output for a given input. So LaTeX Code: f(x) = \\sqrt{x} defines a function LaTeX Code: f which gives the positive square root of a real number LaTeX Code: x . But this function isn't the inverse of the function LaTeX Code: g defined by LaTeX Code: g(x) = x^2 . In fact, if LaTeX Code: g is a function from the real numbers to the real numbers, it has no inverse function.
Something else to watch out for: sometimes the term "square root of x" refers to the positive square root, that is, LaTeX Code: \\sqrt{x} \\equiv x^{1/2} , while other times, the "square root of x" means "a real number, y, such that y2 = x", where y can be positive or negative.
http://en.wikipedia.org/wiki/Square_root
I meant, are there functions that have a different way of processing input?
I am basically wondering if the "+1" answer to my question is ALWAYS right, i.e. none consider anything else correct.
Like 1+1=2.
Sorry for my English...
Rasal, heh don't worry.. I obviously know stuff like that! I always provide the +- y answer unless it would be illogical (like in a practical assignment).
Heh, I am starting to feel slightly stupid after asking this question.. :P
Edit: Yeah, thinking of it as an absolute-value makes sense though this is different from things like sqrt X = +-y (where +-y is the correct answer, only +y is wrong).. I mean, you can get either 1 or -1 depending on how you read it..
Also why is the other way around (eliminating the exponents first) wrong?
I am no expert so this is just for discussion.Some little detail has been giving me a head-ache for 1 of my problems.. And here is why.
Does ((-1)^{2})^{1/2} equal 1 or -1? You decide...
When we say 11/2 we are saying perform the function of raising 1 by the power of (1/2) which is subtly different to asking what are the possible roots of the equation x2=1.
Instead of function use operation.