Determine which statements are true. Check all that apply.

h(x) has a constant output of –2.50.
As x increases, g(x) increases.
g(x) is greater than –2.50 for x values less than –1.
h(x) is less than –2.50 for x values greater than –2.
The input value for which g(x) = h(x) is between –1 and 0. 

To determine which statements are true, let's break down each statement one by one and analyze it based on the behavior of the functions \( g(x) \) and \( h(x) \). I'll assume that you have the functions or their graph available, but since the functions are not explicitly given here, I'll explain how you can check each statement for truth:

### Statement 1: \( h(x) \) has a constant output of –2.50.
- If \( h(x) \) is a constant function with an output of –2.50, this means that for all values of \( x \), the output is always –2.50. 
- **True** if \( h(x) \) is indeed a constant function with a value of –2.50.

### Statement 2: As \( x \) increases, \( g(x) \) increases.
- This refers to the function \( g(x) \) being **increasing**. You would check if \( g(x) \) grows larger as \( x \) increases, meaning the slope is positive.
- **True** if \( g(x) \) is an increasing function as \( x \) increases.

### Statement 3: \( g(x) \) is greater than –2.50 for \( x \) values less than –1.
- For this to be true, the graph or values of \( g(x) \) must be higher than –2.50 when \( x < -1 \).
- **True** if \( g(x) > -2.50 \) for \( x < -1 \).

### Statement 4: \( h(x) \) is less than –2.50 for \( x \) values greater than –2.
- Since \( h(x) \) is said to be a constant at –2.50 (from statement 1), it cannot be less than –2.50 for any values of \( x \), including \( x > -2 \).
- **False** if \( h(x) \) is constant at –2.50, as it won't change and cannot be less than –2.50.

### Statement 5: The input value for which \( g(x) = h(x) \) is between –1 and 0.
- This implies that the two functions \( g(x) \) and \( h(x) \) intersect between \( x = -1 \) and \( x = 0 \). To check this, you'd need to know where their values are equal within this range.
- **True** if there exists an intersection of \( g(x) \) and \( h(x) \) in the interval \( (-1, 0) \).

### Summary:
- The **true** statements are likely:
  1. \( h(x) \) has a constant output of –2.50.
  2. As \( x \) increases, \( g(x) \) increases.
  3. \( g(x) \) is greater than –2.50 for \( x \) values less than –1.
  5. The input value for which \( g(x) = h(x) \) is between –1 and 0.

- The **false** statement is:
  4. \( h(x) \) is less than –2.50 for \( x \) values greater than –2.

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Would you like further explanation of any of the steps or more details?

Here are 5 related questions to consider:
1. How do you determine if a function is increasing or decreasing?
2. What are the conditions for two functions to intersect?
3. How would you graph a constant function like \( h(x) = -2.50 \)?
4. Can a constant function ever be less than its own value at different points?
5. How would you solve \( g(x) = h(x) \) algebraically if the functions were given?

**Tip:** When comparing functions graphically, intersections are points where the functions have the same value at the same input!

Determine which statements are true. Check all that apply.
h(x) has a constant output of –2.50.
As x increases, g(x) increases.
g(x) is greater than –2.50 for x values less than –1.
h(x) is less than –2.50 for x values greater than –2.
The input value for which g(x) = h(x) is between –1 and 0.

Explore how to determine true statements about the behavior of g(x) and h(x) functions, including their increasing behavior, constant values, and points of intersection. This problem involves graph analysis and inequalities, suitable for Grades 9-11.

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