I'm working on finding inverse functions. I came across the following and the result is something I've struggled with for a long time. The function is:[math]f(x) = 2-5x[/math] and for the inverse I got:[math]\frac{x-2}{-5}[/math]However, the answer is given as:[math]\frac{2-x}{5}[/math]I know this is the answer I got multiplied by -1, but why and how can that be legal if we're not also multiplying the other side of the equation by -1?
Transforming an expression
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Dr.Peterson
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Your answer is equivalent to theirs (but not as pretty). You can always multiply both the numerator and the denominator of a fraction by the same number; use -1, and you'll get their form (after distributing and rearranging).
We are multiplying by ONE, not by minus 1. Anything divided by itself (other than zero) equals 1.
[math] \dfrac{x - 2}{-5} = \dfrac{x - 2}{-5} * 1 = \dfrac{x - 2}{-5} * \dfrac{-1}{-1} = \dfrac{-x + 2}{5} = \dfrac{2 - x}{5}. [/math]
Transforming an expression by multiplying it by a convenient form of 1 or by adding to it a convenient form of zero are basic techniques in transforming expressions.
Why is it not as pretty? I always viewed the variable coming first (and as a positive) as being the proper way to write the expression. But I keep seeing it written (2-x)/5 in books. Why is the (2-x)/5 best practice?
Yes, I meant multiply both numerator and denominator by -1. I phrased that poorly. Thank you.
Dr.Peterson
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We generally tend to avoid negative denominators, though they aren't illegal.
Whether the numerator is 2-x or -x+2 is just a matter of taste. Or perhaps context -- I'd prefer either to the other under different circumstances -- depending on whether I dislike negative leading terms, or non-descending order, more.
Also, keep in mind that your answer not looking the same as the book's doesn't mean it's wrong.
The Highlander
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I have the same predilection as you (starting an expression with "the variable", eg: x) but starting an expression with a positive term (rather than a negative one) does take precedence (from an aesthetic point of view) and a negative denominator really does "look" ugly. ?
However, chacun à son goût, does apply and nobody could fault your answer mathematically. ?
Got it. That's very helpful. Thank you very much.
Otis
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Hi jpanknin. You may write the variable first this way:
\(\displaystyle -\frac{x - 2}{5}\quad\)
Once you've multiplied a sufficient number of algebraic ratios by [imath]\frac{-1}{-1}[/imath], you'll come to realize that a negation symbol in a ratio (as in the forms shown below) may be shifted to one of three positions: in the numerator, in the denominator or out in front.
\(\displaystyle \frac{-a}{b} \; = \; \frac{a}{-b} \; = \; -\frac{a}{b}\)
Therefore, when you see something like [imath]\frac{x - 2}{-5}[/imath], you may immediately consider alternate forms:
\(\displaystyle \frac{-(x - 2)}{5} \quad -\frac{x - 2}{5}\)
[imath]\;[/imath]