Question

determine the values for x when the given function is increasing.
options:
$-300 leq x leq 300$
$0 leq x leq 12$
$10 leq x leq 12$
$200 leq x leq 400$
$7 leq x leq 10$

Explanation:

Step1: Understand increasing function

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

Step2: Analyze each interval

• Option \( -300 \leq x \leq 300 \): These are \( y \)-values, not \( x \)-values for increase, so incorrect.
• Option \( 0 \leq x \leq 12 \): The graph doesn't increase the entire time here (e.g., from 5 - 6 it decreases, 7 - 10 it decreases then increases? Wait, no, let's check the graph points. Wait, the key is to find where the graph is rising. Looking at the options, \( 10 \leq x \leq 12 \): From \( x = 10 \) (the minimum point) to \( x = 12 \), the graph rises ( \( y \) increases as \( x \) increases). Let's check other options:
• \( 0 \leq x \leq 12 \): Has decreasing parts (like 5 - 6, 7 - 10).
• \( 200 \leq x \leq 400 \): These are \( y \)-values, not \( x \).
• \( 7 \leq x \leq 10 \): The graph decreases here (from \( x = 7 \) peak to \( x = 10 \) minimum).
• \( 10 \leq x \leq 12 \): From \( x = 10 \) (low point) to \( x = 12 \), the graph goes up ( \( y \) increases as \( x \) increases), so this interval is where the function is increasing.

Answer:

\( 10 \leq x \leq 12 \) (corresponding to the option: \( 10 \leq x \leq 12 \))