Calculus How To

Properties of Functions

Share on

Functions >

A function takes an input (x) and produces a single output (y) for each x-value. Functions also pass the vertical line test; if you draw a vertical line through the graph of a function, it never intersect the graph more than once.

properties of functions

This graph is a function because it passes the vertical line test.

Properties of Functions

All functions have certain properties, or distinct features, which can be very useful when trying to analyze them. The following common properties of functions describe how a graph is shaped, what happens as x-values increase, and whether the functions can be further analyzed with calculus:

  1. Domain and Range
  2. Even or Odd
  3. Increasing or Decreasing
  4. Maxima and Minima
  5. Concave Up or Down
  6. Bounded or Unbounded
  7. Continuous or Discontinuous
  8. Differentiable or Non-Differentiable

1. Domain and Range

The set of all inputs (e.g., x-values) is called the domain. For example, the f(x) = x2 can have any number as an x-value, so the domain is (-∞, ∞). The range is the set of all outputs (e.g., y-values). Watch the video for an overview of the domain and range:

Can’t see the video? Click here.
See also: How to find the domain and range of a function.

2.Even or Odd

  • symmetrical function

    An even function.

    Even functions have symmetry about the vertical axis: f(-x) = x for every x.
  • properties of function odd or even

    An odd function.

    Odd functions have symmetry about the origin: f(-x) = x for every x in the domain.

Functions do not have to have even or odd symmetry: they can be neither.
More info and tests: Even and Odd Functions

3.Increasing or Decreasing

The graph of f(x)=3x (red) is increasing. the graph of f(x) =-3x + 9 (blue) is decreasing.

Increasing functions travel upwards from left to right; as x increases, y decreases. With decreasing functions, as you move from left to right, the graph goes downward. In other words, as x increases, y decreases.

More info and formal definitions: Increasing and Decreasing Functions

4.Maxima and Minima

The lowest point on a graph is called the minima; the highest point on a graph is called the maxima. “Global means “the entire graph of the function” while “local” refers to “a small part of the graph”.

relative minimum

Image Source:


5.Concave Up or Down

Concavity tells you if a graph is shaped like the letter U (concave up) or an upside down U (concave down).

graph of square function 2

Concavity properties of functions: Concave up (red) and concave down (blue).

See: Concave up and down functions.

6. Bounded or Unbounded

Bounded functions have boundaries or constraints placed upon them. For example, if you were looking at a function describing car speed, you would be constrained by the speed limit of the car (perhaps 100 m.p.h.). Unbounded functions don’t have any constraints.
See: Bounded and unbounded functions.

7. Continuous or Discontinuous

Continuous functions have no breaks, jumps or holes. Otherwise, they are discontinuous. In calculus, we are most interested in those functions that are continuous, because discontinuity makes it challenging to analyze them.

Can’t see the video? Click here.
See also:

8. Differentiable or Non-Differentiable

“Differentiable” means there is a slope that you can calculate. This slope will tell you something about the rate of change: how fast or slow an event (like acceleration) is happening.

The derivative must exist for every point in the domain, otherwise the function is not differentiable. This might happen when you have a hole in the graph: if there’s a hole, there’s no slope (there’s a dropoff!).

Can’t see the video? Click here.
See: Differentiable vs. Non Differentiable Functions.


Some graphs created with Desmos.

Stephanie Glen. "Properties of Functions" From Calculus for the rest of us!

Need help with a homework or test question? With Chegg Study, you can get step-by-step solutions to your questions from an expert in the field. Your first 30 minutes with a Chegg tutor is free!

Leave a Reply

Your email address will not be published. Required fields are marked *