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. $f(x) = 3^x$

Explanation:

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

A linear function is in the form \( f(x) = mx + b \), where \( m \) is the slope. For \( f(x) = -\frac{2}{3}x \), the slope \( m = -\frac{2}{3} \). Since the slope is negative (\( m < 0 \)), as \( x \) increases, \( f(x) \) decreases. So this function is decreasing.

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

A constant function has the form \( f(x) = c \), where \( c \) is a constant. Here, \( f(x) = -2.5 \) means the output value is always -2.5, regardless of the input \( x \). So this function is constant.

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

For an exponential function \( f(x) = a^x \), if \( a > 1 \), the function is increasing. Here, \( a = 3 \) which is greater than 1. As \( x \) increases, \( 3^x \) grows larger (e.g., when \( x = 1 \), \( f(1) = 3 \); when \( x = 2 \), \( f(2) = 9 \); when \( x = 3 \), \( f(3) = 27 \), etc.). So this function is increasing.

Answer:

a. \( f(x) = -\frac{2}{3}x \) is a decreasing function.
b. \( f(x) = -2.5 \) is a constant function.
c. \( f(x) = 3^x \) is an increasing function.