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?
Best Answer
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.