Question

determine whether the intervals over which the function is increasing, decreasing, or constant and determine whether the function is odd, even or neither. f(x)= radical (x^2-1)

Answer 1

Are you in Calculus? Do you know how to take derivatives?

There are a couple ways to do this but I need to know how advanced your math class is.

Answer 2

The function is even.
\[\sqrt{(-x) ^{2}-1} = \sqrt{(x) ^{2}-1} \]

Answer 3

To determine whether the interval over the function is increasing, decreasing, or constant, graph the function, either with a graphing utility or by hand (you can graph by hand by just plugging in values for x to find what is the y value).

The answer is decreasing from -infinity to -1, constant from -1 to +1, and increasing from +1 to infinity.

Answer 4

You can find the intervals over which the function is increasing, decreasing or constant, you can use the first derivative test. That's by testing the signs of the derivative. We have \(f'(x)=\frac{x}{\sqrt{x^2-1}}\). The derivative has negative values on \((-\infty,-1)\), positive values at \((1,\infty)\), and it's undefined on \((-1,1)\). Therefore, the function is decreasing on \((-\infty,-1)\) and increasing on \((1,\infty)\).