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Let's analyze the given problem:

We are provided with a function \( f(z) = \frac{z^2 + 8}{z - 2} \), and it is stated that the function is continuous for all real numbers. The task is to find the value of \( f(-2) \).

The function seems to have a discontinuity at \( z = 2 \) because the denominator becomes zero. To ensure continuity, there must be some special handling, likely through factoring or simplifying, to define \( f(z) \) at \( z = 2 \).

The computation already shows:

\[
f(-2) = \frac{(-2)^2 + 8}{-2 - 2} = \frac{4 + 8}{-4} = \frac{12}{-4} = -3.
\]

However, the answer seems to ask for \( f(2) \), but the question involves ensuring the function is continuous everywhere, including at \( z = 2 \).

It appears the answer for \( f(-2) \) isn't zero as indicated. Could you confirm if the equation being evaluated is actually for \( z = -2 \)? The continuity detail might suggest a hole or redefinition at \( z = 2 \).

Let me know if you'd like a deeper dive into continuity handling at \( z = 2 \), or a full simplification analysis.

If a function f is continuous for all real numbers, and if f(z) = (z^2 + 8) / (z - 2) when z ≠ 2, then f(−2) = ?

Learn how to solve a problem involving the continuity of a rational function where f(z) = (z^2 + 8) / (z - 2) and determine the value of f(−2). This problem covers key concepts in calculus, including the continuity of functions and how to deal with limits.

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