Math Problem Statement

If the function f is continuous for all real numbers and if f(x) = (x^2 - 4)/(x + 2) when x ≠ -2, then f(-2) = ?

Solution

To solve this problem, we need to determine the value of f(2)f(-2) given that the function f(x)=x24x+2f(x) = \frac{x^2 - 4}{x + 2} is continuous for all real numbers.

Step-by-Step Solution

  1. Simplify the Expression for f(x)f(x):

    The given function is: f(x)=x24x+2f(x) = \frac{x^2 - 4}{x + 2}

    Notice that the expression is undefined at x=2x = -2 because the denominator becomes zero. To simplify the expression, factor the numerator:

    x24=(x2)(x+2)x^2 - 4 = (x - 2)(x + 2)

    Thus, we can rewrite f(x)f(x) as:

    f(x)=(x2)(x+2)x+2f(x) = \frac{(x - 2)(x + 2)}{x + 2}

    For x2x \neq -2, the x+2x + 2 terms cancel out:

    f(x)=x2(for x2)f(x) = x - 2 \quad \text{(for } x \neq -2\text{)}

  2. Find the Limit as x2x \to -2:

    Since f(x)f(x) must be continuous for all real numbers, including at x=2x = -2, the limit of f(x)f(x) as x2x \to -2 must equal f(2)f(-2). Calculate the limit:

    limx2f(x)=limx2(x2)\lim_{x \to -2} f(x) = \lim_{x \to -2} (x - 2)

    Substitute x=2x = -2:

    limx2(x2)=22=4\lim_{x \to -2} (x - 2) = -2 - 2 = -4

  3. Determine the Value of f(2)f(-2):

    For the function to be continuous at x=2x = -2, we must have:

    f(2)=limx2f(x)=4f(-2) = \lim_{x \to -2} f(x) = -4

Conclusion

The value of f(2)f(-2) is 4-4. Thus, the correct answer is:

A) 4\boxed{\text{A) } -4}

Would you like further details, or do you have any questions?


Here are 5 related questions for further exploration:

  1. How would you define continuity of a function at a point?
  2. What happens if a function is not continuous at a point?
  3. How do you find the limit of a rational function as it approaches a point of discontinuity?
  4. Can a function be continuous but not differentiable at some point?
  5. What is the significance of factoring the numerator and denominator in rational functions?

Tip: Always check for continuity and differentiability separately; continuity does not always imply differentiability!

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Math Problem Analysis

Mathematical Concepts

Continuity
Limits
Factoring
Rational Functions

Formulas

f(x) = (x^2 - 4) / (x + 2)
Factoring: x^2 - 4 = (x - 2)(x + 2)
Limit: lim(x → -2) (x - 2) = -4

Theorems

Continuity of a function
Limit evaluation for rational functions

Suitable Grade Level

Grades 11-12