Question

determine the domain on which the following function is increasing.

Explanation:

Step1: Understand the concept of increasing function

A function \(y = f(x)\) is increasing on an interval if for any two points \(x_1\) and \(x_2\) in that interval with \(x_1eq0\)), if \(a>0\) (opens - up), it is decreasing on the interval \((-\infty,-\frac{b}{2a})\) and increasing on the interval \((-\frac{b}{2a},\infty)\). From the graph, we can also use the visual method: as \(x\) increases, if the \(y\) - value of the function also increases, the function is increasing.

Step2: Locate the vertex and analyze the direction of the parabola

The given parabola opens - up (since it has a minimum point). The vertex of the parabola divides the domain into two intervals. Looking at the graph, the \(x\) - coordinate of the vertex is \(x = 1\).

Step3: Determine the increasing interval

For an open - up parabola, the function is increasing to the right of the vertex. So, when \(x>1\), as \(x\) increases, \(y\) also increases.

Answer:

The function is increasing on the interval \((1,\infty)\)