Question

Determine the open intervals on a given function.

Answer 1

To determine the open intervals on a given function, you need to identify the x-values where the function is defined and then determine where the function is increasing or decreasing.

This can be done by finding the x-values where the function is defined, looking for any vertical asymptotes or holes in the graph, and analyzing the sign of the first derivative to determine the intervals where the function is open.

Step-by-step explanation:

The open intervals on a given function are the intervals where the function is continuous and the graph is not interrupted by any holes or jumps. To determine the open intervals, you need to identify the x-values where the function is defined and then determine where the function is increasing or decreasing.

The open intervals can be found by looking at where the function crosses the x-axis or changes from increasing to decreasing or vice versa.

1. Start by finding the x-values where the function is defined. These are the values that make the denominator of a fraction or the radicand of a square root non-zero. For example, in the function f(x) = 1/x, x cannot be 0.
2. Next, look for any vertical asymptotes or holes in the graph. These are points where the function approaches infinity or is undefined. For example, in the function f(x) = (x+2)/(x-3), x = 3 is a vertical asymptote.
3. Finally, determine the intervals where the function is increasing or decreasing. This can be done by analyzing the sign of the first derivative. If the first derivative is positive, the function is increasing. If the first derivative is negative, the function is decreasing. Use this information to determine the intervals where the function is open (continuous).

For example, let's take the function f(x) = x^2 - 4x + 3. The x-values where the function is defined are all real numbers. The graph of the function does not have any vertical asymptotes or holes.

The first derivative of the function is f'(x) = 2x - 4. Setting this equal to zero, we find x = 2. This means the function is increasing for x < 2 and decreasing for x > 2. Therefore, the open intervals are (-∞, 2) and (2, ∞).