+189 We can find the inverse of an arbitrary linear function and find its inverse and determine how many instances there are when the linear function is an inverse of itself.
\(y = mx + b\) is the typical form of a linear function. Let's find its inverse.
1) Swap the variables x and y.
\(x = my + b\)
2) Solve for y-variable.
\(x = my + b \\ my = x - b \\ y = \frac{1}{m}x - \frac{b}{m}\)
Now we know that for an arbitrary linear equation \(y = mx + b\), \(y = \frac{1}{m}x - \frac{b}{m}\) is the inverse. When are these equations equal? These equations are equal when the coefficient of x and the constant term are the same! This leads to a system of equations.
\(m = \frac{1}{m}; b = -\frac{b}{m} \\ m = \frac{1}{m} \\ m^2 = 1 \\ m = 1 \text{ or } m = -1\)
There are two possibilities for m. Now, let's find the corresponding values for b that make linear functions their own inverse.
\(\text{Let } m = 1: \\ b = -\frac{b}{m} \\ b = -b \\ 2b = 0 \\ b = 0\)
This mean that when m = 1 and b = 0, the linear function is its own inverse. This means that \(y = x\) is its own inverse!
Now, let's try find the other value of b when m = -1.
\(\text{Let } m = -1 \\ b = -\frac{b}{m} \\ b = -\frac{b}{-1} \\ b = b\)
The left-hand side and the right-hand side of the equation are the same equation. This means that the value of b can be anything, and the equation will hold true. In other words, \(y = -x + b\) where b is any real number has the property that the linear function is its own inverse. This means that y = -x and y = -x + 1, y = -x + 3, y = -x + 1000000000000 are all inverses of themselves.
This means there are infinitely many linear functions that have inverses of themselves.
+189 We can find the inverse of an arbitrary linear function and find its inverse and determine how many instances there are when the linear function is an inverse of itself.
\(y = mx + b\) is the typical form of a linear function. Let's find its inverse.
1) Swap the variables x and y.
\(x = my + b\)
2) Solve for y-variable.
\(x = my + b \\ my = x - b \\ y = \frac{1}{m}x - \frac{b}{m}\)
Now we know that for an arbitrary linear equation \(y = mx + b\), \(y = \frac{1}{m}x - \frac{b}{m}\) is the inverse. When are these equations equal? These equations are equal when the coefficient of x and the constant term are the same! This leads to a system of equations.
\(m = \frac{1}{m}; b = -\frac{b}{m} \\ m = \frac{1}{m} \\ m^2 = 1 \\ m = 1 \text{ or } m = -1\)
There are two possibilities for m. Now, let's find the corresponding values for b that make linear functions their own inverse.
\(\text{Let } m = 1: \\ b = -\frac{b}{m} \\ b = -b \\ 2b = 0 \\ b = 0\)
This mean that when m = 1 and b = 0, the linear function is its own inverse. This means that \(y = x\) is its own inverse!
Now, let's try find the other value of b when m = -1.
\(\text{Let } m = -1 \\ b = -\frac{b}{m} \\ b = -\frac{b}{-1} \\ b = b\)
The left-hand side and the right-hand side of the equation are the same equation. This means that the value of b can be anything, and the equation will hold true. In other words, \(y = -x + b\) where b is any real number has the property that the linear function is its own inverse. This means that y = -x and y = -x + 1, y = -x + 3, y = -x + 1000000000000 are all inverses of themselves.
This means there are infinitely many linear functions that have inverses of themselves.
+189 I did not forget about y = x in my answer. In fact, here is a direct quote from my answer contradicting your claim:
"... [W]hen m = 1 and b = 0, the linear function is its own inverse. This means that \(y = x\) is its own inverse!"
Also, the question specifically concerns itself about "how many linear functions" there are with a particular property. Even if I had conveniently missed y = x, the answer would still be infinitely many, as I later concluded in my answer.
Despite your insinuation, I am not AI. I am answering these questions genuinely, and I continue doing my best to generate high-quality answers geared towards the mathematical ability of the asker. If you have spotted a mistake in any of my previous answers, you can tell me. I will fix the mistake and give you credit accordingly for spotting the error.
+170 Well it looks like I missed that in your original answer, but I stand by my "insinuation" that your answer is AI-generated.
+2489 Plaintainmountain, this post, along with your other posts of BS and idiocy, demonstrates a serious inaccuracy in the general assessment that you are an Incompetent Moronic Fool. Your post demonstrates that you are much closer to the idiot side of the imbecile-idiot level, and you are quickly regressing to the complete idiot level.
You see AI when it’s not there. You lack experience and discernment in looking at solutions presented by someone who’s educated and practiced in mathematics, and also educated and practiced in communications and in teaching the subject.
The3Mathketeers presents competent formal instruction math solutions without sounding like a falsely erudite, pedantic teenager. That doesn’t happen by accident. IO hope he stays around for awhile. This forum definitely needs an infusion of (non-artificial) intelligence.
There is/was a member of this forum who seemed to be from another world. https://web2.0calc.com/members/thexsquaredfactor/ I’ve never read presentations like his on any other forum –including restricted access forums. He is/was truly a wonder!
At present, there are no AIs that reproduce formal instruction at this level. Though I suppose it’s only a matter of time before they can. There are a few restricted and paid-subscription AIs that come close, but the AI artifacts are still obvious to anyone familiar with high-level, formal teaching of mathematics and presentations of the solutions.
Most fools can tell the difference between Gold and Fool’s Gold –leastwise they can if they’ve seen enough real gold. But you cannot. You’ve not seen enough real gold, or you are just a fool’s fool.
...Someday, bananas may grow on Plaintainmountain. Bananas are brain food. You should eat more of them....
GA
--. .-
Lol. Plaintainmountain is raging. But seriously he needs to stop assuming accurate answers are ai. There are more users plaintainmountain goes against. IhaveHairyStuff, tastyabananas2ndDad, SpectraSynth, Ginstioniff, and more.
Please do not worry about the (rather inebriated) chimp GingerAle. She parades around the fact that she is “Genetically enhanced” and “a bloody mind reader” as a justification for her repeated violations of social norms. Being a non-organic animal doesn’t make you better than a human – you are still a primate, but with a perverse obsession.
Of course, GA with her high and mighty intelligence doesn’t consider the above facts and looks condescendingly upon anyone with lesser wit; whoever answers any question wrong must be an imbecilic troll or a "Bullshit Bug".
