For all real numbers x, the function f is defined by f(x) = x^2 - k, where k is a constant… — GRE General Test

For all real numbers

x

, the function

f

is defined by

f(x) = x^2 - k

, where

k

is a constant. The custom operation

\star

is defined for all real numbers

a

and

b

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

5 \star (-3) = -12

, what is the value of

k

The value of

k

is 18.

Evaluating

f(2) = 4 - k

f(5) = 25 - k

, and

f(-3) = 9 - k

yields

5 \star (-3) = (4 - k) - (34 - 2k) = -30 + k

. Setting

-30 + k = -12

correctly gives

k = 18

Evaluate

f(a+b)

for

a=5

and

b=-3

a + b = 5 + (-3) = 2

, so

f(2) = 2^2 - k = 4 - k

Substitute the input

a+b = 2

into the function definition

f(x) = x^2 - k

Evaluate

f(a)

and

f(b)

individually

f(5) = 5^2 - k = 25 - k

and

f(-3) = (-3)^2 - k = 9 - k

Apply the function rule to inputs 5 and -3.

Substitute the evaluated expressions into the custom operation definition

5 \star (-3) = (4 - k) - [(25 - k) + (9 - k)] = (4 - k) - (34 - 2k) = -30 + k

Simplify the algebraic expression by combining like terms and distributing the negative sign.

Solve for

k

using the given equation

5 \star (-3) = -12

-30 + k = -12 \implies k = 18

Add 30 to both sides of the equation to isolate

k

Evaluating custom binary operations by substituting function definitions and simplifying algebraic expressions.

Estimated Time:1m 30s

Question