For all non-zero real numbers x, the function f satisfies the equation 2f(x) + f\left(\fra… — GMAT

For all non-zero real numbers $x$ , the function $f$ satisfies the equation $2f(x) + f\left(\frac{1}{x}\right) = 6x$ . What is the value of $f(2)$ ?

Answer: 7

The value of

f(2)

is 7.

Substituting

x = 2

into the functional equation gives

2f(2) + f\left(\frac{1}{2}\right) = 12

. Substituting

x = \frac{1}{2}

gives

2f\left(\frac{1}{2}\right) + f(2) = 3

. Multiplying the first equation by 2 and subtracting the second equation eliminates

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

, producing

3f(2) = 21

, so

f(2) = 7

Substitute

x = 2

into the functional equation

2f(2) + f\left(\frac{1}{2}\right) = 12

This establishes a linear relation between

f(2)

and

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

Substitute

x = \frac{1}{2}

into the functional equation

2f\left(\frac{1}{2}\right) + f(2) = 3

This establishes a second linear relation involving the reciprocal input.

Eliminate

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

by solving the system of equations

f(2) = 7

Multiplying the first equation by 2 yields

4f(2) + 2f\left(\frac{1}{2}\right) = 24

. Subtracting the second equation gives

3f(2) = 21

, which simplifies to

f(2) = 7

Evaluating functional equations by reciprocal variable substitution to form and solve a linear system.

Estimated Time:2m 0s

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