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VCE Stuff => VCE Mathematics => VCE Mathematics/Science/Technology => VCE Subjects + Help => VCE Mathematical Methods CAS => Topic started by: ZachCharge on May 17, 2010, 07:16:45 pm

Title: Sketching x^n where n is a fraction...how?
Post by: ZachCharge on May 17, 2010, 07:16:45 pm: Hail,

So we have a SAC on trig and composite functions Thrusday/Friday this week. Today our teacher sorta slipped out transformations of functions of $x^n$ where n is a fraction. The only problem is, we only touched this in class weeks ago and my notes on it are not up to scratch. Math Quest as expected does not mention these. Anyone have any documentation that explains how to go about these?
Must remain third...must remain third...or higher

Thanks in advance.
Title: Re: Sketching x^n where n is a fraction...how?
Post by: |ll|lll| on May 17, 2010, 07:42:01 pm: What kind of fractions are you looking at?

Basically, if n = 1r2, the graph would be a square root graph.
When n = 1r3, the graph would be a cube root graph and so on.
Try negative fractions on your CAS too...

However, I can't help much because I haven't learnt composite functions yet. ;)
Good luck nevertheless
Title: Re: Sketching x^n where n is a fraction...how?
Post by: cameron_15 on May 17, 2010, 07:49:29 pm: n^1/2 is the square root function, which is half a parabola on its side.

(http://personal.georgiasouthern.edu/~bhlee/info/math_1111/graphs/square_root_big.gif)

n^1/3 is the cube root funtion, which is a cubic graph on its side (sort of).

(http://personal.georgiasouthern.edu/~bhlee/info/math_1111/graphs/cube_root_big.gif)

They should be the only ones you get...

...and very soon someone awesome at maths is going to give you a way better summary thing of this in a few minutes...
Title: Re: Sketching x^n where n is a fraction...how?
Post by: |ll|lll| on May 17, 2010, 07:54:53 pm: ^ Lol yeah, maybe.

For the graph y = x^(1r3), it's actually the graph of the inverse function of y = x^3.
So for y = x^(1r n ), just think of the inverse of the function y = x ^ n
Title: Re: Sketching x^n where n is a fraction...how?
Post by: the.watchman on May 17, 2010, 07:58:02 pm: Quote from: cherylim23 on May 17, 2010, 07:54:53 pm
^ Lol yeah, maybe.

For the graph y = x^(1r3), it's actually the graph of the inverse function of y = x^3.
So for y = x^(1r n ), just think of the inverse of the function y = x ^ n

EXCEPT anything with an even denominator, in which only the positive side is taken :)
And also, fractions with even numerators have both portions to the graph, eg. x^{2/5} and x^{4/3}
Title: Re: Sketching x^n where n is a fraction...how?
Post by: ZachCharge on May 17, 2010, 08:00:47 pm: Thanks for the help, now I don't feel lost. :)
Title: Re: Sketching x^n where n is a fraction...how?
Post by: m@tty on May 17, 2010, 08:24:05 pm: Quote from: cherylim23 on May 17, 2010, 08:07:41 pm
Quote from: the.watchman on May 17, 2010, 07:58:02 pm
EXCEPT anything with an even denominator, in which only the positive side is taken :)
Oh yeah, my bad :P Because it's not defined when you take the even-number root of a negative integer :)

Not defined over $\mathbb{R}$ ..

Also it is worth noting that if n has both an even numerator and denominator, two arms are present, both positive and negative, even(lol) though it simplifies to a function with a normally narrower domain.

Eg $x^{\frac{2}{4}}=x^{\frac{1}{2}}$ . So it is the normal square root graph, BUT it's domain is $\mathbb{R}$ (not $\mathbb{R}^+ \cup \{0\}$ ).
Title: Re: Sketching x^n where n is a fraction...how?
Post by: Mao on May 19, 2010, 12:42:29 am: Quote from: m@tty on May 17, 2010, 08:24:05 pm

Eg $x^{\frac{2}{4}}=x^{\frac{1}{2}}$ . So it is the normal square root graph, BUT it's domain is $\mathbb{R}$ (not $\mathbb{R}^+ \cup \{0\}$ ).

Really? Do you have sources?

Fractional powers are open to interpretation, so do I take the interpretation that $x^{p/q} = \sqrt[q]{x^p}$ or $x^{p/q} = (\sqrt[q]{x})^p$ ? The prior would suggest you are right, the latter would suggest you are wrong.

Or, to generalise, do we always accept $x^{2/2} = |x|$ ?
Title: Re: Sketching x^n where n is a fraction...how?
Post by: naved_s9994 on May 19, 2010, 06:12:57 pm: Quote from: m@tty on May 17, 2010, 08:24:05 pm

Eg $x^{\frac{2}{4}}=x^{\frac{1}{2}}$ . So it is the normal square root graph, BUT it's domain is $\mathbb{R}$ (not $\mathbb{R}^+ \cup \{0\}$ ).

I dont think thats the case either, from memory...
Its in the methods textbook (essentials), somewhere Ch 1-4 : Dont have the book on me atm, otherwise, i would have directly reffered you..