If g(x) is the inverse of f(x), what is the value of f(g(2))?

Asked by maham237 @ 26/12/2020 in Mathematics viewed by 351 People

A. -6
B. -3
C. 2
D. 5

Answered by maham237 @ 26/12/2020

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 brainly.com/question/3225044
The piecewise-defined functions brainly.com/question/9590016
The composite function 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

If g(x) is the inverse of f(x), what is the value of f(g(2))?

Further explanation

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