Decreasing and Positive Function Calculator

A decreasing and positive function is a mathematical function that always produces positive outputs while its value decreases as the input increases. This type of function is common in economics, physics, and other fields where quantities decrease over time while remaining positive.

What is a Decreasing and Positive Function?

A decreasing and positive function is defined as a function f(x) where:

For any two points x₁ and x₂ where x₁ < x₂, f(x₁) > f(x₂)
For all x in the domain, f(x) > 0

These functions are strictly decreasing and always produce positive results. Common examples include exponential decay functions, logarithmic functions, and certain polynomial functions.

Formula and Calculation

The general form of a decreasing and positive function can be represented as:

f(x) = a * e^(-k*x) + c

Where:

a is the initial value (a > 0)
k is the decay constant (k > 0)
c is the asymptote value (c ≥ 0)

This formula represents exponential decay with a positive asymptote. The function will always be positive and will decrease as x increases.

Worked Examples

Example 1: Simple Exponential Decay

Given f(x) = 10 * e^(-0.5*x):

At x = 0: f(0) = 10 * e⁰ = 10
At x = 1: f(1) = 10 * e^-0.5 ≈ 6.065
At x = 2: f(2) = 10 * e^-1 ≈ 3.679

The function decreases while remaining positive for all x ≥ 0.

Example 2: With Asymptote

Given f(x) = 5 * e^(-0.2*x) + 2:

At x = 0: f(0) = 5 + 2 = 7
At x = 5: f(5) = 5 * e^-1 + 2 ≈ 3.679 + 2 ≈ 5.679
As x → ∞, f(x) → 2

The function approaches 2 as x increases but never goes below 2.

Frequently Asked Questions

What makes a function decreasing and positive?

A function is decreasing and positive if its value decreases as the input increases while always producing positive outputs. This requires the function to be strictly decreasing and bounded below by zero.

Can a linear function be decreasing and positive?

Yes, a linear function of the form f(x) = -k*x + c where k > 0 and c > k*x_min can be decreasing and positive. The function must have a positive y-intercept and a negative slope.

What are practical applications of decreasing and positive functions?

These functions model phenomena like radioactive decay, drug concentration in the bloodstream, and depreciation of assets. They're also used in economics to model demand curves and in physics for half-life calculations.