For all real numbers x, the function f is defined by f(x) = x^2 - 2x. The custom binary operation \otimes is defined for all real numbers a and b by a \otimes b = f(a + b) - f(a - b) If k is a constant such that (k \otimes 3) \otimes 2 = 88, what is the value of k?

For all real numbers x, the function f is defined by f(x) = x^2 - 2x. The custom binary op… — GRE General Test

For all real numbers

x

, the function

f

is defined by

f(x) = x^2 - 2x

. The custom binary operation

\otimes

is defined for all real numbers

a

and

b

a \otimes b = f(a + b) - f(a - b)

k

is a constant such that

(k \otimes 3) \otimes 2 = 88

, what is the value of

k

The value of

k

is 2.

Expanding the definition

a \otimes b = f(a+b) - f(a-b)

using

f(x) = x^2 - 2x

gives

[(a+b)^2 - 2(a+b)] - [(a-b)^2 - 2(a-b)] = (a^2 + 2ab + b^2 - 2a - 2b) - (a^2 - 2ab + b^2 - 2a + 2b) = 4ab - 4b = 4b(a-1)

. Evaluating

k \otimes 3

yields

12k - 12

. Substituting this into

(12k - 12) \otimes 2

yields

4(2)(12k - 12 - 1) = 96k - 104

. Setting

96k - 104 = 88

gives

96k = 192

, so

k = 2

Express the custom operation

a \otimes b

in simplified algebraic terms.

a \otimes b = f(a+b) - f(a-b) = [(a+b)^2 - 2(a+b)] - [(a-b)^2 - 2(a-b)] = 4ab - 4b = 4b(a - 1)

Expanding and canceling common terms simplifies the binary operation definition.

Evaluate the inner operation

k \otimes 3

k \otimes 3 = 4(3)(k - 1) = 12k - 12

Substitute

a = k

and

b = 3

into the simplified operation formula

4b(a - 1)

Evaluate the outer operation

(12k - 12) \otimes 2

(12k - 12) \otimes 2 = 4(2)[(12k - 12) - 1] = 8(12k - 13) = 96k - 104

Substitute

a = 12k - 12

and

b = 2

into

4b(a - 1)

Set the resulting expression equal to 88 and solve for

k

96k - 104 = 88 \implies 96k = 192 \implies k = 2

Linear algebraic equation solving yields the value of

k

Custom Binary Operations and Nested Function Evaluation