Use interval notation to write the intervals over which f is (a) increasing, (b) dec and (c) constant.
The Correct Answer and Explanation is :
To determine the intervals over which a function ( f(x) ) is increasing, decreasing, or constant, we analyze its behavior as ( x ) progresses from left to right on its graph. Here’s how to approach this:
- Increasing Intervals: A function is increasing on an interval if, for any two points ( x_1 ) and ( x_2 ) within that interval where ( x_1 < x_2 ), the corresponding function values satisfy ( f(x_1) < f(x_2) ). Graphically, this means the curve ascends as it moves from left to right. In interval notation, if ( f(x) ) increases between ( x = a ) and ( x = b ), we denote this as ( (a, b) ).
- Decreasing Intervals: A function is decreasing on an interval if, for any ( x_1 < x_2 ) within that interval, ( f(x_1) > f(x_2) ). This indicates the graph descends as ( x ) increases. For instance, if ( f(x) ) decreases from ( x = c ) to ( x = d ), we write ( (c, d) ).
- Constant Intervals: A function is constant on an interval if ( f(x_1) = f(x_2) ) for all ( x_1 ) and ( x_2 ) within that interval. This results in a horizontal line segment on the graph, indicating no change in ( f(x) ) as ( x ) varies. If ( f(x) ) remains constant between ( x = e ) and ( x = f ), we express this as ( (e, f) ).
Steps to Identify These Intervals from a Graph:
- Observe the Graph: Starting from the leftmost point, trace the graph to the right, noting where it ascends, descends, or remains flat.
- Identify Critical Points: Points where the graph changes direction (peaks and troughs) or flattens are crucial. These are often local maxima, minima, or points of inflection.
- Determine Intervals: Based on these critical points, segment the ( x )-axis into intervals where the function is consistently increasing, decreasing, or constant.
Example:
Consider a function ( f(x) ) with the following behavior:
- It increases from ( x = -5 ) to ( x = -2 ).
- Remains constant from ( x = -2 ) to ( x = 1 ).
- Increases again from ( x = 1 ) to ( x = 3 ).
- Decreases from ( x = 3 ) to ( x = 5 ).
In interval notation:
- Increasing: ( (-5, -2) \cup (1, 3) )
- Constant: ( (-2, 1) )
- Decreasing: ( (3, 5) )
It’s essential to use open intervals (parentheses) because at the exact points where the function changes direction, it is neither increasing nor decreasing. These points are transitions and don’t belong to any interval of monotonicity.
By carefully analyzing the graph and noting these intervals, we can accurately describe where the function increases, decreases, or remains constant.