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Difficulty: MediumFunctions and Custom Symbol Operations

The function ff is defined for all real numbers xx by f(x)=2x2−xf(x) = 2x^2 - x. For all real numbers aa and bb, the custom operation ⋆\star is defined by a⋆b=f(a+b)−f(a−b)a \star b = f(a + b) - f(a - b). What is the value of 2⋆(−3)2 \star (-3)?

  1. -42Answer
  2. B
    42
  3. C
    -46
  4. D
    -18
  5. E
    78

Answer

The correct answer is -42.
Evaluating a+b=2+(−3)=−1a + b = 2 + (-3) = -1 gives f(−1)=2(−1)2−(−1)=3f(-1) = 2(-1)^2 - (-1) = 3. Evaluating a−b=2−(−3)=5a - b = 2 - (-3) = 5 gives f(5)=2(5)2−5=45f(5) = 2(5)^2 - 5 = 45. Subtracting the two function values yields 3−45=−423 - 45 = -42.

Step-by-Step Solution

1
Identify the values for variables aa and bb and evaluate the inputs for the function ff.
Here a=2a = 2 and b=−3b = -3. Therefore, a+b=2+(−3)=−1a + b = 2 + (-3) = -1 and a−b=2−(−3)=5a - b = 2 - (-3) = 5.
The definition of the custom operation requires evaluating f(a+b)f(a+b) and f(a−b)f(a-b).
2
Calculate f(−1)f(-1) using the function rule f(x)=2x2−xf(x) = 2x^2 - x.
f(−1)=2(−1)2−(−1)=2(1)+1=3f(-1) = 2(-1)^2 - (-1) = 2(1) + 1 = 3.
Substituting x=−1x = -1 into the quadratic expression.
3
Calculate f(5)f(5) using the function rule f(x)=2x2−xf(x) = 2x^2 - x.
f(5)=2(5)2−(5)=2(25)−5=45f(5) = 2(5)^2 - (5) = 2(25) - 5 = 45.
Substituting x=5x = 5 into the quadratic expression.
4
Compute the difference f(a+b)−f(a−b)f(a+b) - f(a-b).
3−45=−423 - 45 = -42.
Evaluating f(−1)−f(5)f(-1) - f(5) according to the definition of ⋆\star.

Key Concept

Evaluating nested algebraic functions and custom symbol operations by direct substitution.
Estimated Time:1m 30s
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Topics

Functions and Custom Symbol Operations

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