Question

If $f\left(\frac{x + 1}{2x - 1}\right) = 2x,\ x \in \mathbb{N}$, then the value of $f(2)$ is equal to:

• A. $1$
• B. $4$
• C. $3$
• D. $2$
• E. $5$

Hint:

When solving functional equations of the type $f(g(x)) = h(x)$, you don't necessarily need to find the general formula for $f(x)$. Solving for a specific $x$ is usually much faster when only a single value like $f(2)$ is requested.

Answer 1

The Correct Option is D

Concept: To find the value of a function $f(k)$, we must determine what value of the input variable $x$ makes the expression inside the function parentheses equal to $k$. Once $x$ is found, we substitute it into the output side of the equation.

Step 1: Find the value of $x$ for which the input is $2$.
Set the expression inside $f$ equal to $2$: \[ \frac{x + 1}{2x - 1} = 2 \] Multiply both sides by $(2x - 1)$ to clear the fraction: \[ x + 1 = 2(2x - 1) \] \[ x + 1 = 4x - 2 \]

Step 2: Solve for $x$.
Rearrange the terms to isolate $x$: \[ 1 + 2 = 4x - x \] \[ 3 = 3x \Rightarrow x = 1 \] Since $x = 1$ is a natural number ($x \in \mathbb{N}$), this is a valid solution.

Step 3: Calculate $f(2)$.
Now substitute $x = 1$ into the output expression $2x$: \[ f(2) = 2(1) = 2 \]