Question

this graph shows a function.
for which intervals is the function increasing? select all that apply.
2 < x < 5
2 < x < 6
6 < x < 9
5 < x < 7
0 < x < 2
5 < x < 6

Explanation:

Step1: Recall increasing function definition

A function is increasing on an interval if as \( x \) increases, \( y \) also increases (the graph rises from left to right).

Step2: Analyze each interval

• \( 2 < x < 5 \): The graph is horizontal (constant \( y = 6 \)), so not increasing.
• \( 2 < x < 6 \): From \( x = 2 \) to \( x = 5 \), it's constant; from \( x = 5 \) to \( x = 6 \), it rises. But since part is constant, the whole interval isn't increasing. Wait, no—wait, at \( x = 2 \) to \( x = 5 \), \( y \) is 6 (constant), then \( x = 5 \) to \( x = 6 \), \( y \) goes from 6 to, say, 7? Wait, no, the graph: at \( x = 2 \), \( y = 6 \); \( x = 5 \), \( y = 6 \); then \( x = 5 \) to \( x = 7 \), it rises to 9. Wait, maybe I misread. Let's re - examine the graph:
• From \( x = 0 \) to \( x = 2 \): The graph goes from \( y = 2 \) to \( y = 6 \), so as \( x \) increases, \( y \) increases. So \( 0 < x < 2 \) is increasing.
• From \( x = 2 \) to \( x = 5 \): \( y = 6 \) (constant), so not increasing.
• From \( x = 5 \) to \( x = 7 \): The graph goes from \( y = 6 \) to \( y = 9 \), so as \( x \) increases, \( y \) increases. So intervals within \( 5 < x < 7 \) (like \( 5 < x < 6 \), \( 5 < x < 7 \)) are increasing. Also, \( 0 < x < 2 \) is increasing. Let's check each option:
• \( 2 < x < 5 \): Constant, so no.
• \( 2 < x < 6 \): Has a constant part, so no.
• \( 6 < x < 9 \): The graph is decreasing (from \( y = 9 \) to \( y = 6 \) as \( x \) goes from 7 to 10, so \( 6 < x < 9 \) is part of decreasing, so no.
• \( 5 < x < 7 \): From \( x = 5 \) ( \( y = 6 \)) to \( x = 7 \) ( \( y = 9 \)), \( y \) increases as \( x \) increases, so this interval is increasing.
• \( 0 < x < 2 \): From \( x = 0 \) ( \( y = 2 \)) to \( x = 2 \) ( \( y = 6 \)), \( y \) increases as \( x \) increases, so this interval is increasing.
• \( 5 < x < 6 \): Part of \( 5 < x < 7 \), since from \( x = 5 \) ( \( y = 6 \)) to \( x = 6 \), \( y \) increases (say, to 7? Wait, the graph at \( x = 5 \) is 6, \( x = 6 \) is, looking at the grid, between \( x = 5 \) and \( x = 7 \), it's a line from (5,6) to (7,9). So at \( x = 6 \), \( y=\frac{6 + 9}{2}=7.5\)? So as \( x \) goes from 5 to 6, \( y \) increases, so \( 5 < x < 6 \) is increasing.
• Wait, let's list correct intervals:
• \( 0 < x < 2 \): Increasing.
• \( 5 < x < 7 \): Increasing (since from \( x = 5 \) to \( x = 7 \), \( y \) goes from 6 to 9).
• \( 5 < x < 6 \): Part of \( 5 < x < 7 \), so increasing.

So the correct intervals are \( 0 < x < 2 \), \( 5 < x < 7 \), \( 5 < x < 6 \). Wait, let's check the options again:

• \( 0 < x < 2 \): Yes, because as \( x \) increases from 0 to 2, \( y \) increases from 2 to 6.
• \( 5 < x < 7 \): Yes, as \( x \) increases from 5 to 7, \( y \) increases from 6 to 9.
• \( 5 < x < 6 \): Yes, as \( x \) increases from 5 to 6, \( y \) increases (from 6 to, say, 7.5).

Answer:

The intervals where the function is increasing are \( 0 < x < 2 \), \( 5 < x < 7 \), \( 5 < x < 6 \) (i.e., the options \( 0 < x < 2 \), \( 5 < x < 7 \), \( 5 < x < 6 \)).