What's the use of curve fitting for a function $y = f(x)$, if I can reduce it to a linear regression? Suppose I have data points $(x_i,y_i)$, say $N$ points. I know they are supposed to fit the curve $y = f(x)$. Are there techniques more advanced than linear regression, for such cases to fit the curve? I am asking this because I can get the data points $(f(x_i), y)$ from $(x_i,y_i)$ and they fit on a line, so I can just use linear regression.
I don't know much statistics or probability.

First, you should determine if there is an expected shape for $y(x)$ based on the particular data you are evaluating. Then, you should estimate the uncertainty of each of the y values, e.g., as standard deviations based on multiple measurements if possible.  Then, plot $y = f(x)$ including uncertainty intervals about the y values.  A valid curve fit should fit within the uncertainty intervals, minimize the least squares measure, and agree with the expected shape, if know the expected shape.  Tools such as Mathematica offer curve fitting algorithms including a polynomial best fit one, called Fit.
