Understanding Increasing Functions: Definition, Examples, and Properties

Increasing function

An increasing function, also known as a strictly increasing function, is a type of function that preserves the order of numbers

An increasing function, also known as a strictly increasing function, is a type of function that preserves the order of numbers. In other words, if you have two numbers a and b such that a < b, then the function f(x) is considered increasing if f(a) < f(b). Mathematically, a function f(x) is said to be increasing if for any two values a and b in the domain of the function such that a < b, the corresponding function values f(a) and f(b) satisfy f(a) < f(b). Let's consider an example to understand this concept better. Suppose we have the function f(x) = 2x, where x is the variable. We want to determine if this function is increasing. To do this, we need to compare the values of f(a) and f(b) for any two numbers a and b such that a < b. Let's consider two arbitrary numbers, a = 2 and b = 4. We can evaluate the function for these values: f(a) = f(2) = 2(2) = 4 f(b) = f(4) = 2(4) = 8 Since f(a) = 4 < 8 = f(b), we can conclude that f(x) = 2x is an increasing function. One important thing to note is that in an increasing function, the difference between f(a) and f(b) is always positive when a < b. This means that as the input values increase, the function values also increase. It is also worth mentioning that a function can be increasing on a specific interval or over the entire domain. For example, a function may be increasing in the interval [0, ∞) but decreasing in the interval (-∞, 0]. In summary, an increasing function is one that preserves the order of numbers, where the function values increase as the input values increase.

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