using the given graph of the function f, find the following.\n(a) the intercepts, if any\n(b) its domain and…

Explanation:

Step1: Find the intercepts

• x - intercepts: The points where the graph crosses the x - axis. From the graph, the x - intercepts are the x - values for which y = 0. Let's assume from the graph, the x - intercepts are x=-2, x = 2.
• y - intercepts: The point where the graph crosses the y - axis. From the graph, when x = 0, y=-1. So the y - intercept is y=-1.

Step2: Determine the domain and range

• Domain: The set of all possible x - values. Looking at the graph, the function is defined for all x - values from - 4 to 4 (inclusive). So the domain is [-4,4].
• Range: The set of all possible y - values. From the graph, the lowest y - value is - 2 and the highest is 2. So the range is [-2,2].

Step3: Find the intervals of increase, decrease and constancy

• Increasing intervals: Intervals where the function is going up as we move from left to right. From the graph, the function is increasing on the intervals (-2,0) and (2,4).
• Decreasing intervals: Intervals where the function is going down as we move from left to right. The function is decreasing on the intervals (-4,-2) and (0,2).
• Constant intervals: There are no intervals where the function is constant.

Step4: Check if the function is even, odd or neither

• Even function: A function is even if f(x)=f(-x) for all x in the domain. The graph of an even function is symmetric about the y - axis.
• Odd function: A function is odd if f(-x)=-f(x) for all x in the domain. The graph of an odd function is symmetric about the origin.
• From the graph, the function is not symmetric about the y - axis or the origin, so it is neither even nor odd.

Answer:

(a) x - intercepts: x=-2, x = 2; y - intercept: y=-1 (b) Domain: [-4,4]; Range: [-2,2] (c) Increasing intervals: (-2,0), (2,4); Decreasing intervals: (-4,-2), (0,2); Constant intervals: None (d) Neither even nor odd