Understanding Decreasing Functions: Explained with Examples and Graphical Representations

Decreasing function

A decreasing function is a mathematical function that satisfies the property that as the input values increase, the corresponding output values decrease

A decreasing function is a mathematical function that satisfies the property that as the input values increase, the corresponding output values decrease.

To formally define a decreasing function, let’s consider a function f that is defined on an interval I. We say that f is decreasing on I if for any two values a and b in I, where a < b, the corresponding output values of f satisfy f(a) > f(b).

In other words, as the input values increase (moving from left to right on the x-axis), the output values decrease (moving downward on the y-axis).

Graphically, a decreasing function can be represented by a curve that slopes downward from left to right. The slope of the curve is negative, indicating the decreasing nature of the function.

Here is an example of a decreasing function:

f(x) = -3x + 5

Let’s check whether this function satisfies the condition of being decreasing. Take any two values a and b, such that a < b. Evaluating the function at these points, we have: f(a) = -3a + 5 f(b) = -3b + 5 Since a < b, we can conclude that -3a > -3b, as multiplying both sides by -3 reverses the inequality. Additionally, since 5 is a constant, we can safely ignore it. So, we have:

-3a > -3b

This confirms that the function f(x) = -3x + 5 is indeed a decreasing function.

It’s important to note that not all functions are decreasing. For example, a increasing function would have the opposite behavior, where as the input values increase, the output values also increase.

In summary, a decreasing function is a function where as the input values increase, the output values decrease. It can be represented by a curve that slopes downward from left to right and has a negative slope.

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