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.
$\endgroup$ 22 Answers
$\begingroup$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.
$\endgroup$ 3 $\begingroup$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.
$\endgroup$ 3More in general
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