Question

The range of the function \(f(x) = \log_5(25 - x^2)\) is

• A. \([0,5]\)
• B. \([0,2)\)
• C. \((0,2)\)
• D. None of these

Hint:

Always find the range of the inner function first, then apply the logarithm. Log functions convert \((0,a]\) into \((-\infty, \log a]\).

Answer 1

The Correct Option is D

To determine the range of the function \(f(x) = \log_5(25 - x^2)\), we must ensure that the expression inside the logarithm is positive, because the logarithm of a non-positive number is undefined in real numbers. 

1. First, consider the domain of the function: \(25 - x^2 > 0\)
2. This simplifies to: \(x^2 < 25\)
3. The inequality \(x^2 < 25\) implies: \(-5 < x < 5\)
4. Thus, the function is defined for \(x \in (-5, 5)\).
5. For the range, consider the function again: \(f(x) = \log_5(25 - x^2)\)
6. As \(x\) approaches the bounds of the domain (\(x \to -5\) or \(x \to 5\)), the expression inside the logarithm approaches 0. Thus, \(f(x)\) tends to \(-\infty\).
7. When \(x = 0\), we have: \(f(0) = \log_5(25) = 2\)
8. Therefore, by considering \(25 - x^2\) approaching from values greater than 0 to 25 (excluding 25), the range of \(f(x)\) is: \((-\infty, 2)\)

Thus, the correct answer is None of these, as <(-∞, 2)> is not provided in the options.