# Positive Function / Negative Function

## What is a Positive Function?

A positive function has function values greater than zero (i.e., f(x) > 0). The domain (inputs) of the function can be negative, but the outputs (y-values) must be greater than zero. In other words, a positive function has values that are positive for all arguments of its domain.

A non-negative function is similar, except that it includes zero in its range.

Graphically, if a function’s output values are all above the x-axis, then the function is positive. Conversely, if the output values are all below the x-axis, then the function is negative. A function can also be positive for certain function intervals. For example, the function f(x) = x3 is positive on the open interval (0, ∞) but negative on the interval (-∞, 0).

A caution: a positive function isn’t necessarily an increasing function (although it can be). The function f(x) = 4x2 + 2, shown on the above graph, is completely above the x-axis, which means it is a positive function. However, notice that it is only increasing for function values on the right-hand side of the vertical axis; the function is decreasing for values to the left of the y-axis. In other words, positive functions can have derivatives that are negative or positive.

A couple of interesting properties:

• A positive function f(x) is log-convex if log f(x) is convex [1].
• A linear combination of positive functions is a positive function.

## What is a Negative Function?

A negative function has values that are all negative (i.e., f(x) < 0). The domain (inputs) of the function can be positive, but every output (y-value) must be less than zero. In other words, a negative function has values that are negative for all arguments of its domain. Graphically, all output (y) values are below the horizontal axis.

## Positive Function and Integrals

The definite integral of a positive function represents area under the graph of the function from a to b.

A positive function is integrable if it is a measurable function and if the integral is less than infinity [2].

## References

Image created with Desmos.com.
[1] Ni, L. Additional Problems-Set 5. Retrieved March 6, 2021 from: https://mathweb.ucsd.edu/~lni/math220/Pre-pr5.pdf
[2] Hunter, J. Chapter 4: Integration. Retrieved March 6, 2022 from: https://www.math.ucdavis.edu/~hunter/measure_theory/measure_notes_ch4.pdf