2
Further explanation
Given:
g(x) is the inverse of f(x)
Question:
What is the value of \(\boxed{ \ f(g(2)) \? \ }\)
The Process:
The formal definition for the inverse of a function:
Let f be a one-to-one and onto function with the domain A and the range B. Then its inverse function \(\boxed{f^{-1}}\) has the domain B and the range A such that \(\boxed{ \ y = f(x) \leftrightarrow f^{-1}(y) = x \ }.\)
By definition the inverse function \(\boxed{f^{-1}}\) undoes what \(\boxed{f}\) does. That is, if we take \(x\), apply \(f\), and the apply \(f^{-1}\), we arrive back at \(x\) where we started. Similarly, \(\boxed{f}\) undoes what \(\boxed{f^{-1}}\) does. That is why \(\boxed{ \ f \ and \ f^{-1} \ }\) are the inverses of each other.
Note: \(\boxed{\boxed{ \ f(f^{-1}(x)) = x \ and \ f^{-1}(f(x)) = x \ }}\)
In other words, \(\boxed{\boxed{ \ f(f^{-1}(a)) = a \ and \ f^{-1}(f(a)) = a \ }}\)
Let's get back to our problem.
Given that \(g(x)\) is the inverse of \(f(x).\) We write as follows:
\(\boxed{ \ g(x) = f^{-1}(x) \ }\)
We prepare the composite function of \(\boxed{ \ (fog)(x) = f(g(x)). \ }\)
Substitute \(\boxed{ \ g(x) = f^{-1}(x) \ }\) into the composite function.
Thus becoming, \(\boxed{ \ f(f^{-1}(x)) \ }\)
We calculate the value of \(\boxed{ \ f(g(2)) \ }\) rewritten to \(\boxed{ \ f(f^{-1}(2)) \ }\)
Therefore we get the answer, i.e., \(\boxed{\boxed{ \ f(f^{-1}(2)) = 2 \ }}.\)
- - - - - - -
Note:
To find the inverse, we use the same procedures that we used for relations. Drawing the reflection with respect to \(\boxed{ \ y=x \ }\) we get the following picture as in the attachment. We can also discover proof of this problem in the attached picture.
Learn more
- The inverse of a function https://brainly.com/question/3225044
- The piecewise-defined functions https://brainly.com/question/9590016
- The composite function https://brainly.com/question/1691598
Keywords: g(x) is the inverse of f(x), what is the value of f(g(2))?, the composite function, one-to-one and onto function, with the domain A and the range B, y = x, the reflection
