Difference between expression and function

In calculus we generally use the term 'function'. However is it equally valid to use the term 'expression' instead?

Like: differentiate the expression, integrate the expression, etc.

Or is there really some difference between the two terms? I searched the web but couldn't get a proper answer.

• Functions given by formulas are expressions. Not all functions are given by expressions (formulas), though. Draw any set of points in the plane that satisfy the vertical line test: that is a function. You could have done this is a very nasty way that is not captured by any expression. But, it may still make sense to talk about the derivative at some points. Feb 28 '18 at 14:32
• how do you know that it can't be captured by any expression?
– X10D
May 7 '20 at 12:55

All functions (at least the formulas represented by them) are expressions, but not all expressions are functions.

Definition: A function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output.

Definition: An algebraic expression is a mathematical phrase that can contain ordinary numbers, variables (like $x$ or $y$) and operators (like add, subtract, multiply, and divide).

Example: $7a+2b+3c$ is clearly an algebraic expression but it does not have to be a function, since the letters $a,b,c$ don't have to be inputs. But if $a$ is an input, then we would have a function: $$f(a)=7a+2b+3c$$ but $b$ and $c$ would be constants.

Example: $f(x)=4x+9$ with $x\in\mathbb{N}$ is a function because it has a set of inputs ($x$) and a set of outputs (numbers of the form $4x+9$). But $4x+9$ is also an expression as it contains numbers, variables and operators.

• Consider the function $f(x)=2x$ and $g(x,y)=2x + 0y$. Are these functions equal? I am asking because I want to know if differentation of a function with a variable that isn't contained in its argument is valid. In the above example does it make sense to differentiate f with respect to $y$? May 12 '20 at 22:34
• Yes. $\partial f/\partial y=0$. May 13 '20 at 7:53
• But $f(x)$ isn't a function of y. So wouldn't be better to say that the derivative is undefined (with respect to y)? May 13 '20 at 10:51

Expressions are "syntactical" objects, i.e. pieces of language.

Functions are (mathematical) objects, i.e. pieces of the world.

A function is described/specified by an expression; the same function can be described by more than one expression.