Understanding the Inverse of a Composite Function: Exploring ( f(g(x)) 3x ) and ( g(x) frac{x}{3} )

In this article, we explore the concept of composite functions and their inverses, specifically using the given functions:

Defining the Functions

Let's define the functions ( f(x) ) and ( g(x) ) as follows:

Function ( f(x) ): ( f(x) 3x ) Function ( g(x) ): ( g(x) frac{x}{3} )

Our goal is to determine whether the composite function ( f(g(x)) ) and its inverse function are meaningful and how they relate to each other.

Step-by-Step Solution: Finding ( f(g(x)) )

To find ( f(g(x)) ), we start by using the given functions and substitute ( g(x) ) into ( f(x) ).

Step 1: Define ( g(x) )

First, we know that:

Function ( g(x) ): ( g(x) frac{x}{3} )

Step 2: Substitute ( g(x) ) into ( f(x) )

Now, let's substitute ( g(x) ) into ( f(x) ):

Function ( f(g(x)) ):

[ f(g(x)) fleft(frac{x}{3}right) ]

Since ( f(x) 3x ), we substitute (frac{x}{3}) for ( x ) in ( f(x) ):

[ fleft(frac{x}{3}right) 3left(frac{x}{3}right) ]

Perform the multiplication:

[ 3 left(frac{x}{3}right) x ]

Determining the Inverse

From the above step, we have:

[ f(g(x)) x ]

This means that the composite function ( f(g(x)) ) is equal to the identity function, which is simply ( x ).

Understanding the Inverse

Since ( f(g(x)) x ), it implies that ( g(x) ) is the inverse of ( f(x) ) and vice versa. Therefore, the inverse function of ( f(g(x)) ) is simply:

Inverse Function: ( (f(g(x)))^{-1} f(g(x)) )

Hence, the inverse of ( f(g(x)) ) is ( f(g(x)) ) itself, which also returns ( x ) when applied.

Verification Through Swapping Variables

To verify, let's go through the detailed steps:

1. Let ( y frac{x}{3} ).

Solving for ( x ), we get:

[ x 3y ]

Swap ( y ) with ( x ):

[ y 3x ]

2. Now find the inverse of ( f(x) ):

[ f(x) 3x ]

Swap variables:

[ y 3x ]

Solve for ( x ):

[ x frac{y}{3} ]

Swap ( y ) with ( x ) to get:

[ y frac{x}{3} ]

3. Finally, find ( f(g(x)) ):

Substitute ( x/3 ) where ( x ) is in the equation ( y 3x ):

[ f(g(x)) 3left(frac{x}{3}right) ]

Since the 3's cancel out, we have:

[ f(g(x)) x ]

Conclusion

This confirms that the composite function ( f(g(x)) ) is the identity function, and its inverse is exactly the same function:

[ (f(g(x)))^{-1} f(g(x)) x ]

Thus, the inverse of ( f(g(x)) ) is itself, which returns ( x ).