Define the strictly increasing function and strictly decreasing function on an interval.

A function is considered strictly increasing on an interval if, for any two points \( x_1 \) and \( x_2 \) within that interval, where \( x_1 < x_2 \), the function value at \( x_1 \) is less than the function value at \( x_2 \). In simpler terms, as you move from left to right along the interval, the function's output consistently rises.

Characteristics of Strictly Increasing Functions

• For every \( x_1 < x_2 \), it holds that \( f(x_1) < f(x_2) \).
• The graph of the function slopes upwards as you move from left to right.
• There are no flat sections or decreases in the function's values.

Conversely, a function is termed strictly decreasing on an interval if, for any two points \( x_1 \) and \( x_2 \) in that interval, where \( x_1 < x_2 \), the function value at \( x_1 \) is greater than the function value at \( x_2 \). This means that as you progress from left to right, the function's output consistently falls.

Characteristics of Strictly Decreasing Functions

• For every \( x_1 < x_2 \), it holds that \( f(x_1) > f(x_2) \).
• The graph of the function slopes downwards as you move from left to right.
• There are no flat sections or increases in the function's values.

In summary, strictly increasing functions rise without exception, while strictly decreasing functions fall consistently throughout their defined intervals.