Understanding the Composition of Functions with an Example

In this article, we will explore the concept of function composition, specifically focusing on the given function g(x) 1 - frac{x}{1 - x}. We will go through the process of finding the composition of this function, which is often denoted as (g circ frac{1}{g})(x). This involves substituting (frac{1}{g(x)}) into the function g(x). Let's delve into this step-by-step.

Function Composition Basics

Function composition is a mathematical operation where the output of one function is used as the input of another. Given two functions g(x) and h(x), the composition of these functions is defined as:

((g circ h)(x) g(h(x)))

This notation means that we first apply the function h to the input (x), and then use the resulting value as the input to the function g.

The Given Function

g(x) 1 - frac{x}{1 - x}

Let's begin by understanding the given function:

g(x) 1 - frac{x}{1 - x}

This function takes an input (x), subtracts (frac{x}{1 - x}) from it, and returns the result. To ensure the clarity of our work, we should use proper notation. Here, we will denote the function as:

g(x) 1 - frac{x}{1 - x}

Computing the Composition (g circ frac{1}{g})(x)

Step 1: Determine g(frac{1}{g(x)})

First, we need to find (gleft(frac{1}{g(x)}right)). We start by substituting (frac{1}{g(x)}) into the function:

(gleft(frac{1}{g(x)}right) 1 - frac{frac{1}{g(x)}}{1 - frac{1}{g(x)}})

To simplify, we need to understand what (g(x)) is in its simplest form:

(g(x) 1 - frac{x}{1 - x})

Now, let's express (frac{1}{g(x)}):

(frac{1}{g(x)} frac{1}{1 - frac{x}{1 - x}} frac{1}{frac{(1 - x) - x}{1 - x}} frac{1}{frac{1 - x - x}{1 - x}} frac{1}{frac{1 - 2x}{1 - x}} frac{1 - x}{1 - 2x})

Now, substitute (frac{1 - x}{1 - 2x}) back into the function:

(gleft(frac{1 - x}{1 - 2x}right) 1 - frac{frac{1 - x}{1 - 2x}}{1 - frac{1 - x}{1 - 2x}})

Let's simplify this step-by-step:

(1 - frac{frac{1 - x}{1 - 2x}}{1 - frac{1 - x}{1 - 2x}} 1 - frac{1 - x}{1 - 2x} times frac{1 - 2x}{1 - 2x - (1 - x)})

( 1 - frac{1 - x}{1 - 2x - 1 x} 1 - frac{1 - x}{-1 x} 1 - frac{1 - x}{x - 1})

( 1 - frac{1 - x}{x - 1} 1 - (-1) 1 1 2)

Therefore, (gleft(frac{1}{g(x)}right) 2).

Step 2: Apply the Result

Now that we have (gleft(frac{1}{g(x)}right) 2), we need to determine the value of ((g circ frac{1}{g})(x)).

The function composition ((g circ frac{1}{g})(x)) is defined as:

((g circ frac{1}{g})(x) gleft(frac{1}{g(x)}right) 2)

Thus, the composition of the function (g(x)) with (frac{1}{g(x)}) yields the constant value of 2 when evaluated at (x).

Conclusion

Through the process of function composition, we have found that ((g circ frac{1}{g})(x)) equals 2, which contradicts the answer given in the book. It is important to ensure proper notation and denotations to avoid confusion. In future problems involving function composition, always ensure that the operations are well-defined and the notation is clear.

References

[1] "Understanding Function Composition: A Comprehensive Guide" - MathisFun (October 2023)

[2] "Comprehensive Guide to Mathematical Functions" - Global Math Academy (August 2023)