For all non-zero real numbers x, the function f is defined by f(x) = \frac{x^2 - 4}{2x}, a… — GMAT

For all non-zero real numbers $x$ , the function $f$ is defined by $f(x) = \frac{x^2 - 4}{2x}$ , and the function $g$ is defined by $g(x) = 3x - 1$ . What is the positive value of $a$ for which $f(g(a)) = 0$ ?

Answer: 1

Answer

The positive value of

a

for which

f(g(a)) = 0

is 1.

To solve

f(g(a)) = 0

, first find the values of

y

where

f(y) = 0

. The numerator of

f(y) = \frac{y^2 - 4}{2y}

equals zero when

y^2 - 4 = 0

, yielding

y = 2

and

y = -2

. Setting

g(a) = 3a - 1

equal to these roots gives

3a - 1 = 2 \implies a = 1

and

3a - 1 = -2 \implies a = -1/3

. Since

a

must be positive, the correct value is 1.

Step-by-Step Solution

Determine the values of the argument

y

that satisfy

f(y) = 0

y = 2

y = -2

A fraction equals zero when its numerator is zero and its denominator is non-zero. Setting

y^2 - 4 = 0

gives

y = \pm 2

Substitute

g(a) = 3a - 1

into

y

to solve for

a

Solving

3a - 1 = 2

yields

a = 1

; solving

3a - 1 = -2

yields

a = -1/3

Setting the expression for

g(a)

equal to each root of

f(y) = 0

identifies all potential values for

a

Select the value of

a

matching the positivity condition.

a = 1

The question explicitly specifies finding the positive value of

a

Key Concept

Function composition and evaluation of nested functional equations.

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Step-by-Step Solution

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