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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?

Also, if someone write the following equation

$y=y(t) = t^2$

where $y$ represents the dependent variable and t represents the independent variable, then is it wrong to say that $y(t)$ is merely used to represent the value of the dependent variable, $y$ when $t$ equal some value and $y$ in $y(t)$ does not represent a function?

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  • $\begingroup$ Yes, it is wrong. Analysts will come here and tell you about context or whatever, those will just be euphemisms to the real truth: it is wrong. $\endgroup$ – Git Gud Sep 15 '13 at 9:39
  • $\begingroup$ you can, but make sure you don't get in a muddle because you cannot distinguish y and y() $\endgroup$ – Willemien Sep 15 '13 at 10:51
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    $\begingroup$ This is a problematic area. Despite functions playing a central role in much of mathematics there isn't a clear unambiguous notation that mathematicians agree on. I think a good modern view is that $y$ is a function and $y(t)$ is the value of the function evaluated at $t$ in which case it (usually) makes no sense to say $y=y(t)$. But others take the view that $y=y(t)$ is a special use of notation that emphasises that $y$ is a variable that depends on $t$ and so is perfectly acceptable. $\endgroup$ – Dan Piponi Aug 4 '14 at 20:25
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The difference between $y$ and $y(t)$ is this: $y$ is a function and $y(t)$ is the value of the function at the point $t$. Often people write $y = y(t)$ which is supposed to emphasise that $y$ is a function of the variable $t$, but in absolute terms, the two things aren't equal (even if $y$ is a constant function).

Another possible source of confusion is a statement of the form "consider the function $y(t) = t^2$". As mentioned above, $y(t)$ is not a function. However, a function is determined entirely by its domain and its values at each point in the domain. Assuming we know the domain (which people often neglect to state), by specifying $y(t)$ as a formula in $t$, we know the value of the function at each point in its domain which determines the function $y$. So even though neither side of the equation $y(t) = t^2$ is a function, provided the domain is clear, it does uniquely determine a function.

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