Question

determine the domain on which the following function is decreasing.

Explanation:

Step1: Understand the concept of a decreasing function

A function \(y = f(x)\) is decreasing on an interval if for any two points \(x_1\) and \(x_2\) in that interval with \(x_1

Step2: Analyze the graph

Looking at the given graph:

• We first find the vertex (the highest point of the parabola - like graph). The \(x -\) coordinate of the vertex is \(x = 3\).
• For \(x<3\), as \(x\) increases, the \(y -\) value of the function increases (the function is increasing).
• For \(x>3\), as \(x\) increases, the \(y -\) value of the function decreases.
• The function also has an \(x -\) intercept at \(x = 8\). Since the function is only defined (has non - negative \(y\) values in the context of the graph shown) from \(x=- 1\) (where \(y = 48\)) to \(x = 8\) (where \(y = 0\)).
• Combining the analysis of the slope of the graph (using the definition of a decreasing function), we see that for \(x\in(3,8)\), the function is decreasing.

Answer:

The function is decreasing on the interval $(3,8)$.