If I am a conic section, then my e = ∞
Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.
If I am a conic section, then my e = ∞
Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.
Mahan Ghobadi
Maths Tutor- ESL (80)| 2 Unit maths (96)(2013) | 3 Unit maths (99)| 4 Unit maths(95)| Physics (88)| music1(93)
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Mahan Ghobadi
Maths Tutor- ESL (80)| 2 Unit maths (96)(2013) | 3 Unit maths (99)| 4 Unit maths(95)| Physics (88)| music1(93)
Get more answers for your questions, as well as weekly tips and blog posts, from my friends and I at:
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I presently cannot think of a way to determine the value using only Extension 1 Methods (there is the subtle distinction of principal valued cube roots vs generalised cube roots) as there is a necessary step in deducing the magnitude of the complex cube roots from the arguments...
If I am a conic section, then my e = ∞
Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.
Last one does feel like a cheat though.
Last edited by Lugia101101; 3 Dec 2016 at 11:07 PM. Reason: Formatting
This is probably what the question wants, then:
For a function to be invertable, it must be defined such that it is one-to-one. If a function is monotone, then every value will result in a single value of , however, the function must be strictly monotonic, else there could exist two values and such that . So if the function is one-to-one, then there exists an inverse, and hence will only have an inverse if it is strictly monotone.
A function with no two points and such that . An example would be:
Which fits the definition for monotonic increasing, but isn't strictly monotonic increasing. (From Wikipedia)
A function can still have at some point and have an inverse, such as , which is strictly monotonic increasing, has at , and has the inverse function .
Last edited by Lugia101101; 7 Dec 2016 at 10:05 PM. Reason: Mistake Correction
Also, we don't need a function to be monotonic for it to be invertible, since we can easily form discontinuous functions that are not monotonic but pass the horizontal line test. However, you can prove as an exercise that a continuous one-to-one function must be monotonic.
Such as a function defined as ?
Was basically the point of the question. In addition, the function has to be continuous.
However, I haven't heard of the "strict monotonicity" grammar though
The tan function defined above is continuous but not monotonic.
It's easy to come up with a discontinuous function that's not monotonic but is invertible. It suffices to take a graph that has two branches, one starting at the point (0, 2) and decreasing strictly, smoothly and asymptotically to y = 1, and the other branch a reflection of this about the y-axis and shifted down enough so that the overall graph passes the horizontal line test (and only include one of the points at the discontinuity at x = 0).
Last edited by InteGrand; 8 Dec 2016 at 1:23 AM.
Q4 is a trick question
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