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VCE Stuff => VCE Mathematics => VCE Mathematics/Science/Technology => VCE Subjects + Help => VCE Mathematical Methods CAS => Topic started by: kenhung123 on March 22, 2010, 11:03:59 pm

Title: Inverse and reciprocal contradictions...
Post by: kenhung123 on March 22, 2010, 11:03:59 pm: Ok so $2^{-1}=1/2=0.5$ Is that inverse or reciprocal?
If reciprocal then what is $f^{-1}(x)$ ? $1/f(x)$ ?
Title: Re: Inverse and reciprocal contradictions...
Post by: TrueTears on March 22, 2010, 11:05:27 pm: the reciprocal of 2 is 2^-1
Title: Re: Inverse and reciprocal contradictions...
Post by: vexx on March 22, 2010, 11:07:43 pm: Quote from: kenhung123 on March 22, 2010, 11:03:59 pm
Ok so $2^{-1}=1/2=0.5$ Is that inverse or reciprocal?
If reciprocal then what is $f^{-1}(x)$ ? $1/f(x)$ ?

i hope i don't misinterpret your question but the reciprocal is the 'multiplicative inverse', whereby you flip a number, so 1/10, the reciprocal is 10.
whereas an inverse function f^-1, is when you switch 'x' and 'y' and solve for 'y'. they are quite different
Title: Re: Inverse and reciprocal contradictions...
Post by: kyzoo on March 22, 2010, 11:12:11 pm: Reciprocal function is not the same thing as the inverse function, learned this from doing an iTute Methods exam.
Title: Re: Inverse and reciprocal contradictions...
Post by: m@tty on March 22, 2010, 11:18:39 pm: The definition of the inverse function is $f[f^{-1}(x)]=f^{-1}[f(x)]=x$ i.e. if you put a number through one of the functions, then through the other function, you end up where you started, they 'cancel each other out'.

The reciprocal is just, as vexx said, the multiplicative inverse, it is defined such that $a^{-1}ab=aa^{-1}b=b$ , so here again they 'cancel out'.
"In many contexts in mathematics an inverse is defined as the opposite of something" -- sourced from wikipedia -- so here there is no contradiction; merely a case of confusing definitions.
Title: Re: Inverse and reciprocal contradictions...
Post by: /0 on March 22, 2010, 11:27:33 pm: The multiplicative inverse and the reciprocal of 2 are both the same thing, $2^{-1}$ . Reciprocal is just a fancy name for multiplicative inverse.

- The multiplicative inverse of a number $a \neq 0$ is a number of $b$ such that $ab = 1$ . The standard notation for the multiplicative inverse is $\frac{1}{a}$ .
So, for example, $\frac{1}{f(x)}$ is the multiplicative inverse of $f(x)$ (for values of x such that $f(x) \neq 0$ ) because $\frac{1}{f(x)}f(x) = 1$ .

- The inverse function of $f(x)$ is a function $g(x)$ such that $g \circ f(x) = x$ . The standard notation used for the inverse function is $f^{-1}(x)$ .

The different notations indicate different 'types' of inverses.
Title: Re: Inverse and reciprocal contradictions...
Post by: shinny on March 22, 2010, 11:29:44 pm: In trying to simplify things for a VCE level (since I'm barely following these 'mathematically correct' definitions), the reciprocal function is $f(x)^{-1}=\frac{1}{f(x)}$ while the inverse is $f^{-1}(x)$ . When working with inverses, the first step you should do is $let f(x)=y$ to heavily simplify things, because what the inverse means is that you basically swap every x for a y, and every y for an x. So if $f(x)=y=x^2+2$ , the inverse would be $x=y^2+2$ . From here, just use algebra to rearrange for y. If you didn't swap f(x) out for y, you'd end up with some pretty messy (and algebraically incorrect?) working out. I'm aware this might not be completely correct but it's as far as you really need to understand for Methods at least.

EDIT: If you want to think of this graphically, inverse functions as I stated above are formed by swapping x and y. Hence, draw a graph on a cleanly ruled and labelled cartesian axis with a marker, grab the bottom right corner of the sheet of paper and flip it upward so that it becomes the top left (you should be looking at the other side of the piece of paper; hold it up to light if you can't see your graph through on the other side). Ta-da, that's what your inverse function will look like graphically, because what you've effectively done is swap the positions of the x and y axes as you should be able to see.
Title: Re: Inverse and reciprocal contradictions...
Post by: crayolé on March 22, 2010, 11:41:41 pm: Why do i remember reciprocal as putting 1 over the number then making it negative? :/ Is that called something else? Is it only used for perpendicular lines or something?
Title: Re: Inverse and reciprocal contradictions...
Post by: shinny on March 22, 2010, 11:52:54 pm: Quote from: Tezza on March 22, 2010, 11:41:41 pm
Why do i remember reciprocal as putting 1 over the number then making it negative? :/ Is that called something else? Is it only used for perpendicular lines or something?

$m_{1} m_{2}=-1$
$m_{2}=- \frac{1}{m_{1}}$

Yep, perpendicular linear graphs. Anywho, back to topic guys. Perhaps it's best to wait 'til the OP replies before spamming him with more explanations.
Title: Re: Inverse and reciprocal contradictions...
Post by: kenhung123 on March 23, 2010, 07:36:56 am: Alright so reciprocal is a type of inverse and $f^{-1}(x)$ is different from $f(x)^{-1}$

Thanks so much for clearing this up
Title: Re: Inverse and reciprocal contradictions...
Post by: m@tty on March 23, 2010, 08:48:34 am: Yes.