2
Further explanation
Given:
g(x) is the inverse of f(x)
Question:
What is the value of [tex] \boxed{ \ f(g(2)) \? \ } [/tex]
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 [tex] \boxed{f^{-1}} [/tex] has the domain B and the range A such that [tex]\boxed{ \ y = f(x) \leftrightarrow f^{-1}(y) = x \ }.[/tex]
By definition the inverse function [tex] \boxed{f^{-1}} [/tex] undoes what [tex] \boxed{f} [/tex] does. That is, if we take [tex]x[/tex], apply [tex]f[/tex], and the apply [tex] f^{-1} [/tex], we arrive back at [tex]x[/tex] where we started. Similarly, [tex] \boxed{f} [/tex] undoes what [tex] \boxed{f^{-1}} [/tex] does. That is why [tex] \boxed{ \ f \ and \ f^{-1} \ } [/tex] are the inverses of each other.
Note: [tex]\boxed{\boxed{ \ f(f^{-1}(x)) = x \ and \ f^{-1}(f(x)) = x \ }}[/tex]
In other words, [tex]\boxed{\boxed{ \ f(f^{-1}(a)) = a \ and \ f^{-1}(f(a)) = a \ }}[/tex]
Let's get back to our problem.
Given that [tex] g(x) [/tex] is the inverse of [tex] f(x). [/tex] We write as follows:
[tex]\boxed{ \ g(x) = f^{-1}(x) \ }[/tex]
We prepare the composite function of [tex] \boxed{ \ (fog)(x) = f(g(x)). \ } [/tex]
Substitute [tex]\boxed{ \ g(x) = f^{-1}(x) \ }[/tex] into the composite function.
Thus becoming, [tex] \boxed{ \ f(f^{-1}(x)) \ } [/tex]
We calculate the value of [tex] \boxed{ \ f(g(2)) \ } [/tex] rewritten to [tex] \boxed{ \ f(f^{-1}(2)) \ } [/tex]
Therefore we get the answer, i.e., [tex] \boxed{\boxed{ \ f(f^{-1}(2)) = 2 \ }}. [/tex]
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Note:
To find the inverse, we use the same procedures that we used for relations. Drawing the reflection with respect to [tex]\boxed{ \ y=x \ }[/tex] 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
