Question

Find the domain of "f(x) = log(2x - x²)".

Answer 1

The domain of the function f(x) = log(2x - x²) is (-∞, 0) ∪ (2, ∞)

Step-by-step explanation:

The domain of the function f(x) = log(2x - x²) is determined by the values of x that make the logarithm function valid. In this case, the argument of the logarithm must be positive, so we set 2x - x² > 0 and solve for x.

1. We rearrange the inequality: 2x - x² > 0
2. We factor out a common x: x(2 - x) > 0
3. We set each factor equal to zero: x = 0 or 2 - x = 0
4. We find the critical values and create a number line to determine the intervals that satisfy the inequality. Since the inequality is strict, we use open circles to represent the critical values.
5. We test a value in each interval and determine the sign of the inequality. For example:
6. Finally, we write the domain in interval notation: (-∞, 0) ∪ (2, ∞)