Key Concepts and Formulas

• Function Composition: For functions $f$ and $g$, the composition $(f \circ g)(x)$ is defined as $f(g(x))$. When composing multiple functions, like $(f_2 \circ J \circ f_1)(x)$, it is evaluated from right to left: $f_2(J(f_1(x)))$.
• Algebraic Manipulation: Solving for an unknown function often involves isolating it through algebraic operations, including finding common denominators and simplifying fractions.
• Variable Substitution: To transform an expression like $J(g(x))$ into $J(x)$, we can set $t = g(x)$ and then express $x$ in terms of $t$ (i.e., $x = g^{-1}(t)$), substituting these into the expression.

Step-by-Step Solution

We are given three functions:

1. $f_1(x) = \frac{1}{x}$
2. $f_2(x) = 1 - x$
3. $f_3(x) = \frac{1}{1 - x}$

We are also given the functional equation: $(f_2 \circ J \circ f_1)(x) = f_3(x)$ where $x \in R - \{0, 1\}$. Our goal is to find the expression for $J(x)$.

Step 1: Expand the function composition. The given equation $(f_2 \circ J \circ f_1)(x) = f_3(x)$ means that $f_2(J(f_1(x))) = f_3(x)$. Explanation: This step breaks down the composite function into a sequence of function applications, starting from the innermost function.

Step 2: Substitute the expression for $f_1(x)$. Substitute $f_1(x) = \frac{1}{x}$ into the equation: $f_2\left(J\left(\frac{1}{x}\right)\right) = f_3(x)$ Explanation: We replace $f_1(x)$ with its definition to simplify the argument of the $J$ function.

Step 3: Substitute the expression for $f_3(x)$. Substitute $f_3(x) = \frac{1}{1-x}$ into the equation: $f_2\left(J\left(\frac{1}{x}\right)\right) = \frac{1}{1 - x}$ Explanation: We replace $f_3(x)$ with its definition to have the right-hand side of the equation fully defined in terms of $x$.

Step 4: Apply the definition of $f_2(x)$ to the left side. The function $f_2(y) = 1 - y$. In our equation, the input to $f_2$ is $J\left(\frac{1}{x}\right)$. Therefore, applying $f_2$ gives: $1 - J\left(\frac{1}{x}\right) = \frac{1}{1 - x}$ Explanation: We use the definition of $f_2$ by substituting its argument, $J\left(\frac{1}{x}\right)$, into the expression $1-x$.

Step 5: Isolate $J\left(\frac{1}{x}\right)$. Rearrange the equation to solve for $J\left(\frac{1}{x}\right)$: $J\left(\frac{1}{x}\right) = 1 - \frac{1}{1 - x}$ Explanation: We move $J\left(\frac{1}{x}\right)$ to one side and the other terms to the other side to isolate the expression containing $J$.

Step 6: Simplify the expression for $J\left(\frac{1}{x}\right)$. Combine the terms on the right-hand side by finding a common denominator: $J\left(\frac{1}{x}\right) = \frac{1 \cdot (1 - x)}{1 - x} - \frac{1}{1 - x}$ $J\left(\frac{1}{x}\right) = \frac{1 - x - 1}{1 - x}$ $J\left(\frac{1}{x}\right) = \frac{-x}{1 - x}$ This can also be written as: $J\left(\frac{1}{x}\right) = \frac{x}{x - 1}$ Explanation: Performing algebraic simplification is crucial to obtain a manageable form of the expression for $J\left(\frac{1}{x}\right)$.

Step 7: Use substitution to find $J(x)$. We have an expression for $J\left(\frac{1}{x}\right)$. To find $J(x)$, let $t = \frac{1}{x}$. From this substitution, we can express $x$ in terms of $t$: $x = \frac{1}{t}$. Now, substitute $t$ for $\frac{1}{x}$ and $\frac{1}{t}$ for $x$ into the simplified expression for $J\left(\frac{1}{x}\right)$: $J(t) = \frac{\frac{1}{t}}{\frac{1}{t} - 1}$ Explanation: This substitution technique allows us to change the argument of the function $J$ from $\frac{1}{x}$ to $t$, and then we will replace $t$ with $x$ to get the desired $J(x)$.

Step 8: Simplify the expression for $J(t)$. Simplify the complex fraction: $J(t) = \frac{\frac{1}{t}}{\frac{1 - t}{t}}$ $J(t) = \frac{1}{t} \cdot \frac{t}{1 - t}$ $J(t) = \frac{1}{1 - t}$ Explanation: We simplify the fraction by multiplying the numerator by the reciprocal of the denominator.

Step 9: Replace the dummy variable. Replace the dummy variable $t$ with $x$ to obtain the expression for $J(x)$: $J(x) = \frac{1}{1 - x}$ Explanation: Since $t$ was just a placeholder variable, we can replace it with $x$ to express the function $J$ in its standard form.

Step 10: Compare with the given options. We found that $J(x) = \frac{1}{1 - x}$. Let's look at the given functions:

• $f_1(x) = \frac{1}{x}$
• $f_2(x) = 1 - x$
• $f_3(x) = \frac{1}{1 - x}$

Our result $J(x) = \frac{1}{1 - x}$ is exactly the definition of $f_3(x)$.

Therefore, $J(x) = f_3(x)$.

Common Mistakes & Tips

• Order of Operations: Always apply function compositions from right to left. For $(f \circ g \circ h)(x)$, evaluate $h(x)$ first, then $g$ of that result, and finally $f$ of that result.
• Algebraic Errors: Be meticulous with algebraic manipulations, especially with signs and common denominators. An error in simplification can lead to an incorrect final function.
• Variable Substitution: When using substitution like $t = g(x)$, remember to also express the original variable ($x$) in terms of the new variable ($t$) and substitute it into the entire expression.

Summary

The problem required finding an unknown function $J(x)$ given a composite functional equation. By systematically expanding the composition, substituting the known function definitions, and performing algebraic manipulations to isolate $J\left(\frac{1}{x}\right)$, we obtained an expression for $J\left(\frac{1}{x}\right)$. A variable substitution ($t = \frac{1}{x}$) was then used to transform this expression into $J(x)$. The final result, $J(x) = \frac{1}{1 - x}$, was identified as $f_3(x)$.

The final answer is \boxed{f 3 (x)} which corresponds to option (D).