inverse function problem

letsgetaway

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Jul 16, 2006
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I need help on the following problem. I'm not sure what's the first step I should take. I think I could figure it out from there though. So, please, no answers. :) I just need an idea what to do first.

Code:
Assume the domains of f and f^-1 are (- infinity, infinity).  Solve for x given that

[f^-1 (11x + 15) = -1, f(-1) = 5]
 
letsgetaway said:
I need help on the following problem. I'm not sure what's the first step I should take. I think I could figure it out from there though. So, please, no answers. :) I just need an idea what to do first.

Code:
Assume the domains of f and f^-1 are (- infinity, infinity).  Solve for x given that

[f^-1 (11x + 15) = -1, f(-1) = 5]
This is testing your understanding of the definition of an inverse function. Evaluate both sides of the first equation using f. Then on the left, use the definition of the inverse. On the right, use the second equation.
 
I didn't quite understand what you were saying. I gave it a try anyways. The answer I came up with is 9/5. Something's telling me I might be wrong.
 
letsgetaway said:
I didn't quite understand what you were saying. I gave it a try anyways. The answer I came up with is 9/5. Something's telling me I might be wrong.
Maybe "Evaluate both sides of the first equation using f" is not the right words. How about "Apply the function f to both sides of the first equation?"

By that I mean the following. We are given

\(\displaystyle \L f^{-1} (11x + 15) = -1,\ f(-1) = 5.\)

Apply \(\displaystyle \L f\) to both sides of the first equation means

\(\displaystyle \L f(f^{-1} (11x + 15)) = f(-1).\)

Now use the definition of inverse on the left side. This means use that by definition of the inverse \(\displaystyle \L f^{-1}\), \(\displaystyle \L f(f^{-1}(y)) = y\) holds for any \(\displaystyle \L y\), so

\(\displaystyle \L f(f^{-1} (11x + 15)) = 11x + 15.\)

Use the second equation on the right side:

\(\displaystyle \L f(-1) = 5.\)

Combining the left and right sides

\(\displaystyle \L 11x + 15 = 5.\)

Now solve for \(\displaystyle \L x.\)
 
JakeD said:
letsgetaway said:
I didn't quite understand what you were saying. I gave it a try anyways. The answer I came up with is 9/5. Something's telling me I might be wrong.
Maybe "Evaluate both sides of the first equation using f" is not the right words. How about "Apply the function f to both sides of the first equation?"

By that I mean the following. We are given

\(\displaystyle \L f^{-1} (11x + 15) = -1,\ f(-1) = 5.\)

Apply \(\displaystyle \L f\) to both sides of the first equation means

\(\displaystyle \L f(f^{-1} (11x + 15)) = f(-1).\)

Now use the definition of inverse on the left side. This means use that by definition of the inverse \(\displaystyle \L f^{-1}\), \(\displaystyle \L f(f^{-1}(y)) = y\) holds for any \(\displaystyle \L y\), so

\(\displaystyle \L f(f^{-1} (11x + 15)) = 11x + 15.\)

Use the second equation on the right side:

\(\displaystyle \L f(-1) = 5.\)

Combining the left and right sides

\(\displaystyle \L 11x + 15 = 5.\)

Now solve for \(\displaystyle \L x.\)

Thanks again, Jake! That has helped me a great deal. :D I'm glad I have another problem like this so I can work em out myself.
 
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