For all real numbers x except 0 and 1, the function f(x) satisfies the relation f(x) + 2f\… — GMAT

For all real numbers $x$ except $0$ and $1$ , the function $f(x)$ satisfies the relation $f(x) + 2f\left(\frac{1}{1-x}\right) = 9x$ . What is the value of $f(2)$ ?

Answer: 6

Answer

The value of

f(2)

6

Evaluating the functional equation

f(x) + 2f\left(\frac{1}{1-x}\right) = 9x

at the values

x = 2

x = -1

, and

x = \frac{1}{2}

creates a system of three linear equations in terms of

f(2)

f(-1)

, and

f\left(\frac{1}{2}\right)

. Solving this system yields

9f(2) = 54

, which simplifies directly to

f(2) = 6

Step-by-Step Solution

Evaluate the functional equation at

x = 2

f(2) + 2f\left(\frac{1}{1-2}\right) = 9(2) \implies f(2) + 2f(-1) = 18

Applying the input

x = 2

creates an equation connecting

f(2)

and

f(-1)

Evaluate the functional equation at

x = -1

f(-1) + 2f\left(\frac{1}{1-(-1)}\right) = 9(-1) \implies f(-1) + 2f\left(\frac{1}{2}\right) = -9

Evaluating at the new input

x = -1

generates a second equation connecting

f(-1)

and

f\left(\frac{1}{2}\right)

Evaluate the functional equation at

x = \frac{1}{2}

f\left(\frac{1}{2}\right) + 2f\left(\frac{1}{1-1/2}\right) = 9\left(\frac{1}{2}\right) \implies f\left(\frac{1}{2}\right) + 2f(2) = 4.5

Evaluating at

x = \frac{1}{2}

completes the cyclic chain by linking back to

f(2)

Solve the system of three linear equations for

f(2)

From equation 3,

f\left(\frac{1}{2}\right) = 4.5 - 2f(2)

. Substituting into equation 2 yields

f(-1) = 4f(2) - 18

. Substituting into equation 1 gives

f(2) + 2(4f(2) - 18) = 18 \implies 9f(2) = 54 \implies f(2) = 6

Eliminating

f\left(\frac{1}{2}\right)

and

f(-1)

isolated

f(2)

to find its exact numerical value.

Key Concept

Cyclic Functional Equations and Substitution Systems

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