Inverse Functions: Given f(x) = 6x + 5, find f^(-1) o f^(-1) o f^(-1) (x)

[rxchelm]

Hi all,

I had a question regarding inverse functions and combining them: I'm afraid I'm not entirely sure what the topic is exactly called so I'm very sorry if this is misleading. In my textbook, it's under inverse functions, and that's it.

But anyways, the question is as follows:

. . . . .\(\displaystyle \mbox{Given }\, f(x)\, =\, 6x\, +\, 5,\, \mbox{ find }\, f^{-1}\, \circ\, f^{-1}\, \circ\, f^{-1}\,(x)\)

I've come up with an answer, but can't find a solution, and so I don't know if I have it right. First, I found the inverse of the function. That would be y = (x-5)/6. Then, I plugged that into the question. Here's my work:

\(\displaystyle f^{-1}\, \circ\, f^{-1}\, \circ\, f^{-1}\,(x)\)

. . . . .\(\displaystyle =\, f^{-1}\, \circ\, f^{-1}\left(\dfrac{x\, -\, 5}{6}\right)\)

. . . . .\(\displaystyle =\,f^{-1}\, \circ\, \left(\dfrac{\left(\dfrac{x\, -\, 5}{6}\right)\, -\, 5}{6}\right)\)

. . . . .\(\displaystyle =\,f^{-1}\, \circ\, \left(\dfrac{x\, -\, 35}{36}\right)\)

. . . . .\(\displaystyle =\, \dfrac{\left(\dfrac{x\, -\, 35}{36}\right)\, -\, 5}{6}\)

. . . . .\(\displaystyle =\,\dfrac{x\, -\, 215}{216}\)

Since math isn't really my strong point, and considering the weirdness of the number, I'm not too sure I got it right.

I'd really appreciate any and all help.

Thanks,
Rachel

 

[stapel]

I'm afraid I'm not entirely sure what the topic is exactly called so I'm very sorry if this is misleading.

The "circle" notation indicates the composition of functions (that is, plugging one function into another), and works in the manner you've shown. (here)

But anyways, the question is as follows:

. . . . .\(\displaystyle \mbox{Given }\, f(x)\, =\, 6x\, +\, 5,\, \mbox{ find }\, f^{-1}\, \circ\, f^{-1}\, \circ\, f^{-1}\,(x)\)

I've come up with an answer, but can't find a solution...[/quote]
In what manner do you have "an answer" but not "a solution"? :shock:

First, I found the inverse of the function. That would be y = (x-5)/6.

Correct.

Then, I plugged that into the question. Here's my work:

\(\displaystyle f^{-1}\, \circ\, f^{-1}\, \circ\, f^{-1}\,(x)\)

. . . . .\(\displaystyle =\, f^{-1}\, \circ\, f^{-1}\left(\dfrac{x\, -\, 5}{6}\right)\)

. . . . .\(\displaystyle =\,f^{-1}\, \circ\, \left(\dfrac{\left(\dfrac{x\, -\, 5}{6}\right)\, -\, 5}{6}\right)\)

. . . . .\(\displaystyle =\,f^{-1}\, \circ\, \left(\dfrac{x\, -\, 35}{36}\right)\)

. . . . .\(\displaystyle =\, \dfrac{\left(\dfrac{x\, -\, 35}{36}\right)\, -\, 5}{6}\)

. . . . .\(\displaystyle =\,\dfrac{x\, -\, 215}{216}\)

Your steps are correct, as is the final form. I'm not sure what you mean by the result being "weird", but fractions would be expected in this sort of exercise.

 

[rxchelm]

What I meant by not finding a solution is that the textbook didn't have solutions for the question, so I had no idea if I'd done it right.

But thank you very much for all your help!