Understanding Increasing Functions: Definition, Examples, and Graphical Representation

increasing function (growth)

An increasing function, also known as a growth function, is a mathematical function that consistently increases as its input values increase

An increasing function, also known as a growth function, is a mathematical function that consistently increases as its input values increase. In other words, as the independent variable of the function increases, the corresponding dependent variable also increases.

To understand this concept, let’s consider a simple example. Suppose we have a function f(x) = 2x, where x is the independent variable and f(x) is the dependent variable. The function states that for every value of x, the corresponding value of f(x) is obtained by multiplying x by 2.

If we plug in some values for x, such as x = 1, x = 2, and x = 3, we can see how the function behaves:

f(1) = 2(1) = 2
f(2) = 2(2) = 4
f(3) = 2(3) = 6

As we can see, as the values of x increase (from 1 to 2 and then to 3), the corresponding values of f(x) also increase (from 2 to 4 and then to 6). This demonstrates that the function is increasing or growing.

In mathematical terms, a function f(x) is increasing on an interval if for any two numbers a and b in that interval, if a < b, then f(a) < f(b). In our example, if a = 1 and b = 3, we have f(1) = 2 and f(3) = 6. Since 1 < 3 and 2 < 6, we can conclude that the function is increasing over the entire interval. Graphically, an increasing function is represented by a curve that rises as you move from left to right on a coordinate plane. The slope of the curve is positive, indicating the upward trend of the function. It's important to note that not all functions are increasing. Some functions may be decreasing (where the function values decrease as the input values increase) or they may be constant (where the function values remain the same regardless of the input values). In summary, an increasing function, or growth function, is a function that consistently increases as its input values increase. It can be identified by analyzing its values and their relationship as well as by examining its graphical representation.

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