Question

1. classify each function as increasing, decreasing, or constant. explain your reasoning.

a. $f(x) = -\frac{2}{3}x$
b. $f(x) = -2.5$
c. $f(x) = 3x$

Explanation:

Part a: Analyze \( f(x) = -\frac{2}{3}x \)

Step 1: Identify the slope

The function is in the form \( f(x) = mx + b \) (linear function), where \( m = -\frac{2}{3} \) and \( b = 0 \).

Step 2: Determine the trend

For a linear function \( f(x)=mx + b \), if \( m<0 \), as \( x \) increases, \( f(x) \) decreases (since the slope is negative, the line falls from left to right). So \( f(x)=-\frac{2}{3}x \) is a decreasing function.

Part b: Analyze \( f(x) = -2.5 \)

Step 1: Recognize the function type

This is a constant function (it can be written as \( f(x)=0x - 2.5 \)). The value of \( f(x) \) does not change regardless of the value of \( x \).

Step 2: Confirm the trend

For any two values \( x_1 \) and \( x_2 \) (where \( x_1
eq x_2 \)), \( f(x_1)=f(x_2)= - 2.5 \). So the function is constant.

Part c: Analyze \( f(x) = 3x \)

Step 1: Identify the slope

The function is linear with \( m = 3 \) (and \( b = 0 \)).

Step 2: Determine the trend

For a linear function \( f(x)=mx + b \), if \( m>0 \), as \( x \) increases, \( f(x) \) increases (the line rises from left to right). So \( f(x) = 3x \) is an increasing function.

Answer:

• a. \( f(x)=-\frac{2}{3}x \) is decreasing (slope \( m = -\frac{2}{3}<0 \), so as \( x \) increases, \( f(x) \) decreases).
• b. \( f(x)= - 2.5 \) is constant (the output value does not change with \( x \)).
• c. \( f(x)=3x \) is increasing (slope \( m = 3>0 \), so as \( x \) increases, \( f(x) \) increases).