Jacob C. answered • 07/23/21
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Adaptive Math and Physics Tutor
You have the equation correct, namely f(x) = √(1 - x2). What you need to do is specify the desired domain. The function f(x) restricted to the domain [0, 1] will yield the unit circle in quadrant 1.
Technically, f(x) = 1/4 * (|x|/x + 1) * (1 - |x - 1|/(x - 1)) * √(1 - x2) will yield f(x) = 0 for x ∈ (-∞, 0) ∪ (1, ∞) and will yield f(x) = √(1 - x2) for x ∈ (0, 1). It will be undefined for x = 0 or x = 1 but at least it doesn't require a domain restriction.
Henry D.
I don't understand the derivation of the equation
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07/23/21
Jacob C.
Henry, what I proposed is not something that you are responsible for knowing in your class. It is something that I’ve discovered on my own and I’m not sure where it falls in the spectrum of mathematics. Consider the expression |x|/x. This expression will be equal to -1 or 1 depending on the sign of x (it’s often called sgn(x) and has a connection to the Heaviside step function). Since the range of the function is {-1, 1}, if we add 1 to the function and divide the result by 2, we shift the range to {0, 1}, which can be recognized as a binary range of sorts. Thus, the function can be used as a sort of “on/off” switch, centered on the desired input. I know that you wish to shut off any outputs generated when x < 0 and when x > 1, so I applied a translation to the switching functions involved and multiplied the two together to obtain the desired effect. If you’d like to dig deeper, please message me directly or send me an email.
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07/24/21
Henry D.
Can you please tell me how you got the equation: f(x) = 1/4 * (|x|/x + 1) * (1 - |x - 1|/(x - 1)) * √(1 - x2)07/23/21