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Zorluk: OrtaFunction Definitions, Evaluation, and Custom Operators

For all real numbers aa and bb, the custom binary operator ⋆\star is defined by a⋆b=a2−b2a \star b = a^2 - b^2. The function ff is defined by f(x)=x⋆3f(x) = x \star 3. If f(x)=4x−12f(x) = 4x - 12, what is the sum of all real values of xx that satisfy this equation?

  1. 4Cevap
  2. B
    3
  3. C
    -4
  4. D
    -3
  5. E
    12

Cevap

The sum of all real values of x that satisfy the equation is 4.
Applying the custom operator definition gives f(x) = x^2 - 9. Setting this equal to 4x - 12 produces the quadratic equation x^2 - 4x + 3 = 0. Factoring yields (x - 1)(x - 3) = 0, which has solutions x = 1 and x = 3. The sum of these solutions is 1 + 3 = 4.

Adım Adım Çözüm

1
Evaluate the custom operator expression f(x) = x * 3.
f(x) = x^2 - 3^2 = x^2 - 9
Applying the definition of the custom operator a * b = a^2 - b^2 with a = x and b = 3.
2
Set f(x) equal to 4x - 12 and rearrange into standard quadratic form.
x^2 - 9 = 4x - 12 => x^2 - 4x + 3 = 0
Equating the function expression to the given linear expression.
3
Factor the quadratic equation to find the roots.
(x - 1)(x - 3) = 0 => x = 1 or x = 3
Finding the values of x that satisfy the quadratic equation.
4
Calculate the sum of the roots.
1 + 3 = 4
Summing all real solutions to satisfy the prompt requirement.

Anahtar Kavram

Custom Operator Evaluation and Quadratic Equation Solving
Tahmini Süre:1m 30s
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Konular

Function Definitions, Evaluation, and Custom Operators

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