Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Is it wrong to represent a dependent variable and a function using the same symbol? For example, can we write the parametric equations of a curve in xy-plane as $x=x(t)$ , $y=y(t)$ where $t$ is the parameter? For me function is different from dependent variable. But, in many calculus texts, sometimes, a dependent variable and a function are represented using the same symbol, why?

share|improve this question
1  
Technically, they are different, but writing this way is an acceptable "abuse of notation" because it saves on notational clutter. –  Prahlad Vaidyanathan Sep 15 '13 at 20:10
    
Abuse of notation. Sometimes that's a good thing, sometimes bad. Different people have different opinions on when it's good and when bad. –  Daniel Fischer Sep 15 '13 at 20:10
    
1  
If it's wrong then I don't wanna be right. –  James S. Cook Sep 15 '13 at 20:13
    
While $y=f(x)$ is a notation that conveys the verbal statement "$y$ is a $f$(unction) of $x$", sometimes the notation $y=y(x)$ is used in order to stress the fact that $y$ is a function of $x$. –  Alecos Papadopoulos Sep 15 '13 at 20:55

1 Answer 1

When we write $y=f(x)$, we give names to the function, its input, and its output.

But not all of these names are needed at all times.

  • "let $f$ be a continuous function" names the function, but neither its input nor output.
  • "$\langle\cdot ,\cdot \rangle$ is jointly continuous" is another version of the above.
  • "$x\mapsto \sqrt{x}$ is an increasing function" names the input, but neither the output nor the function. A less correct, but common version of this statement omits the "$x\mapsto $" part.

What if we want to give names to the input and output, but not to the function itself? Writing $y=\cdot (x)$ is typographically awkward. Writing $y=y(x)$ conveys as much information, and is less likely to be mistaken for a typo. I think "$x\mapsto y$" would be a better choice, but this is less common.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.