2
Further explanation
Given:
g(x) is the inverse of f(x)
Question:
What is the value of ![\boxed{ \ f(g(2)) \? \ }](/tpl/images/0446/1017/ab145.png)
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
has the domain B and the range A such that ![\boxed{ \ y = f(x) \leftrightarrow f^{-1}(y) = x \ }.](/tpl/images/0446/1017/7bbc7.png)
By definition the inverse function
undoes what
does. That is, if we take
, apply
, and the apply
, we arrive back at
where we started. Similarly,
undoes what
does. That is why
are the inverses of each other.
Note: ![\boxed{\boxed{ \ f(f^{-1}(x)) = x \ and \ f^{-1}(f(x)) = x \ }}](/tpl/images/0446/1017/f24f4.png)
In other words, ![\boxed{\boxed{ \ f(f^{-1}(a)) = a \ and \ f^{-1}(f(a)) = a \ }}](/tpl/images/0446/1017/7d0c7.png)
Let's get back to our problem.
Given that
is the inverse of
We write as follows:
![\boxed{ \ g(x) = f^{-1}(x) \ }](/tpl/images/0446/1017/2cbb4.png)
We prepare the composite function of ![\boxed{ \ (fog)(x) = f(g(x)). \ }](/tpl/images/0446/1017/8c3fc.png)
Substitute
into the composite function.
Thus becoming, ![\boxed{ \ f(f^{-1}(x)) \ }](/tpl/images/0446/1017/9476f.png)
We calculate the value of
rewritten to ![\boxed{ \ f(f^{-1}(2)) \ }](/tpl/images/0446/1017/af0e4.png)
Therefore we get the answer, i.e., ![\boxed{\boxed{ \ f(f^{-1}(2)) = 2 \ }}.](/tpl/images/0446/1017/dbf24.png)
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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
we get the following picture as in the attachment. We can also discover proof of this problem in the attached picture.
Learn moreThe inverse of a function link
The piecewise-defined functions link
The composite function link
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
![If g(x) is the inverse of f(x), what is the value of f(g( a. -6 b. -3 c. 2 d. 5](/tpl/images/0446/1017/847ee.jpg)