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An increasing function is one where the function output grows as the input increases. For example, if you imagine walking up a hill, as you walk further, the ground level, representing the function value, goes up.
Mathematically, a function \( f(x) \) is increasing if for any two numbers \( a \) and \( b \), with \( a < b \), it follows that \( f(a) < f(b) \). Understanding this concept is vital when analyzing the behavior of two functions. If both functions \( f \) and \( g \) are increasing:

• \( f \) grows as you move from left to right along the x-axis.
• \( g \) also grows in the same direction.

Imagine if you select \( f(x) = x + 1 \) and \( g(x) = x^2 + 1 \) as your functions, each value of \( f \) and \( g \) increases as \( x \) increases. However, \( g \) grows faster than \( f \), leading to different behaviors in their quotient \( \frac{f}{g} \).
This is a crucial insight when exploring the ratio of two increasing functions.

A decreasing function is the opposite of an increasing function. Here, the function values drop as you move along the input values.
Think about descending a hill: the further you go, the lower you get.
Mathematically, a function \( f(x) \) is decreasing if for any \( a < b \), it holds that \( f(a) > f(b) \).In the context of the exercise, when dealing with the ratio of two functions \( \frac{f}{g} \), if this ratio is decreasing, it suggests that \( f \) grows slower than \( g \).

• Take once more \( f(x) = x + 1 \), and \( g(x) = x^2 + 1 \), where \( g(x) \) grows much rapidly than \( f(x) \).
• This characteristic ensures the quotient \( \frac{f}{g} \) diminishes as \( x \) increases, making the ratio decreasing.

The idea of decreasing functions applies in various scenarios, helping us understand the behavior of complex relationships, such as the ratio of two increasing functions.

The derivative plays a crucial role in measuring how a function changes as its input changes. Think of the derivative as the instantaneous rate of change or the slope of the tangent line to the function curve at any point. When we compute the derivative of a function, it tells us how fast and in which direction the function is moving at that point. In the context of the exercise, we used the derivative of the quotient \( \frac{f}{g} \) to determine if the ratio is increasing or decreasing.

• The derivative is found using the Quotient Rule: \( \left(\frac{f}{g}\right)' = \frac{g' \cdot f - f' \cdot g}{g^2} \).
• If this derivative results in a positive value, the function is increasing; if negative, it's decreasing.

For instance, verifying that \( \frac{x+1}{x^2+1} \) is decreasing involves showing that its derivative is negative across its domain. Calculating derivatives gives us powerful insights into how two functions interact, particularly when looking at their quotients.