Question

determine the domain on which the following function is increasing.

Explanation:

Step1: Understand Increasing Function

A function is increasing when as \( x \) increases, \( y \) also increases. Visually, this is where the graph rises from left to right.

Step2: Identify the Interval

The graph starts at \( x = 0 \) (with a closed dot) and rises until it reaches its peak. From the graph, the peak seems to be around \( x = 5 \) or \( x = 6 \)? Wait, no, looking at the x - axis, the graph starts at \( (0, 55) \) and increases until it reaches the vertex, then decreases. Wait, actually, looking at the x - values: the leftmost point is \( x = 0 \) (since the dot is at \( x = 0 \)) and the graph increases until it reaches the maximum point, then decreases. Wait, the domain of increase is from the left endpoint to the vertex's x - coordinate. Wait, the graph starts at \( x = 0 \) (closed dot) and goes up until \( x = 5 \) or \( x = 6 \)? Wait, no, looking at the x - axis labels: the x - axis has 0,1,2,3,4,5,6,7,8,9,10,11,12,13,14. Wait, the graph starts at \( x = 0 \) (the y - axis) and increases until \( x = 5 \) (or maybe \( x = 6 \))? Wait, no, actually, the function is a parabola - like curve opening downward? Wait, no, the graph starts at \( (0, 55) \), increases to a maximum, then decreases to \( (14, 0) \). Wait, the domain of increase is from \( x = 0 \) to the x - coordinate of the vertex. Wait, looking at the graph, the leftmost point is \( x = 0 \) (since the dot is at \( x = 0 \)) and the graph is increasing when moving from left to right until the peak. So the interval where the function is increasing is from \( x = 0 \) to \( x = 5 \) (or maybe \( x = 5 \) is the vertex? Wait, the x - axis: the grid lines, each square is 1 unit. So the function starts at \( (0, 55) \), goes up, reaches a maximum around \( x = 5 \) (since at \( x = 5 \), it's the highest point before decreasing), then decreases to \( (14, 0) \). Wait, actually, the domain of increase is \( [0, 5] \)? Wait, no, maybe \( [0, 5] \) or \( [0, 5] \)? Wait, let's check the x - values. The function is increasing when \( x \) is in the interval from the left endpoint (x = 0) to the x - coordinate of the maximum point. So the domain on which the function is increasing is \( 0 \leq x \leq 5 \) (or \( [0, 5] \))? Wait, no, maybe I made a mistake. Wait, the graph starts at \( x = 0 \) (the y - axis, since the dot is at \( x = 0 \)) and increases until it reaches the vertex, then decreases. So the interval where the function is increasing is from \( x = 0 \) to \( x = 5 \) (assuming the vertex is at x = 5). Wait, but maybe the vertex is at x = 5 or x = 6? Wait, looking at the graph, the peak is around x = 5 (since at x = 5, the y - value is around 80, then it starts to decrease). So the function is increasing on the interval \( [0, 5] \)? Wait, no, maybe the correct interval is \( [0, 5] \) or \( [0, 5] \). Wait, actually, the leftmost point is x = 0 (closed dot) and the graph is increasing when moving from left to right until the maximum. So the domain of increase is \( 0 \leq x \leq 5 \) (or in interval notation, \( [0, 5] \))? Wait, no, maybe I misread. Wait, the x - axis: the function starts at (0, 55) and goes up, so the interval where it's increasing is from x = 0 to x = 5 (the x - coordinate of the vertex). So the domain on which the function is increasing is \( [0, 5] \)? Wait, no, maybe the correct answer is \( [0, 5] \) or \( 0 \leq x \leq 5 \). Wait, let's confirm: a function is increasing when its derivative (slope) is positive. For a graph, we look at the direction: as x increases, y increases. So from x = 0 (the starting point) to x = 5 (the peak), the graph is rising. So the domain is \( 0 \leq x \leq 5 \) (or in interval notation, \( [0, 5] \)). Wait, but maybe the peak is at x = 5, so the interval is \( [0, 5] \).

Answer:

The function is increasing on the domain \( \boldsymbol{[0, 5]} \) (or \( 0 \leq x \leq 5 \))