Consider the functions f(x) = \frac{x+3}{x-1}, where x \neq 1, and g(x) = x^2 - x - 4. If … — ACT

Consider the functions $f(x) = \frac{x+3}{x-1}$ , where $x \neq 1$ , and $g(x) = x^2 - x - 4$ . If $x$ is an integer such that the composite function evaluation $g(f(g(x))) = 16$ , what is the product of all such integer values of $x$ ?

Answer: -6

Answer

The product of all integer values of

x

that satisfy the equation is

-6

Solving the nested composite equation

g(f(g(x))) = 16

systematically yields the integer solutions

x = 3

and

x = -2

. The product of these values is

-6

Step-by-Step Solution

Set

u = f(g(x))

and solve the outer quadratic equation

g(u) = 16

u = 5

u = -4

This simplifies the nested composition into a single-variable quadratic equation.

Set

v = g(x)

and solve the rational equations

f(v) = 5

and

f(v) = -4

v = 2

v = 0.2

This determines the required outputs of the function

g(x)

that will satisfy the composite equation.

Solve the quadratic equations

g(x) = 2

and

g(x) = 0.2

for

x

x = 3

x = -2

, or

x = \frac{5 \pm \sqrt{445}}{10}

This finds all real values of

x

that satisfy the composite equation.

Identify the integer values from the solution set and compute their product.

3 \times (-2) = -6

The question specifically requests the product of the integer values of

x

Key Concept

Function composition and multi-step equation solving

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