Label function based x coordinate of a given function

This is the purpose of declare function. That is, you can declare a function via

declare function={f(x)=(x)^2-1;}

and then use it e.g. in

node[label={[font=footnotesize]left:{$f(x) =x^2$}}] at (-1.5,{f(-1.5)}) {};

which now knows the value of the function at x=-1.5, or in

draw[color=red,thick] plot (x,{f(x)});

which plots the function. (It is better to use an odd number of samples for symmetric plots, and 100 samples is rather generous, you may want to reduce the sample number and use the smooth key.)

Full code:

documentclass[letterpaper,11pt]{article}
usepackage{tikz}
usetikzlibrary{arrows.meta}

begin{document}
begin{tikzpicture}[domain=-2:2,samples=101,declare function={f(x)=(x)^2-1;}]
    draw[font=small,-Stealth] (-3,0)  -- (3,0) node[right]{$x$} ;
    draw[font=small,-Stealth] (0,-2)  -- (0,2) node[above]{$y$};
        foreach x in {-1,0,...,1}
            draw (x, -2pt) -- ++ (0,4pt);%[shift={(2,1.5)}]
        foreach y in {-1,0,...,1}
            draw (-2pt, y) -- ++ (4pt, 0);%[shift={(2,1.5)}]
    node[label={[font=footnotesize]left:{$f(x) =x^2$}}] at (-1.5,{f(-1.5)}) {};
    clip[rounded corners=10pt] (-3,-2) rectangle (3,2);
    draw[color=red,thick] plot (x,{f(x)});
end{tikzpicture}
end{document}

Note that in pgfplots 1-dimensional plots (of this type) are paths, so you can add labels by specifying the position along the path.

documentclass[letterpaper,11pt]{article}
usepackage{pgfplots}
pgfplotsset{compat=1.17}

begin{document}
begin{tikzpicture}[declare function={f(x)=(x)^2-1;}]
 begin{axis}[axis lines=middle,ymax=2.2,ymin=-1.2,
    xlabel={$x$},ylabel={$y$}]
  addplot[color=red,domain=-2:2,smooth,semithick] {f(x)}
    node[pos=0.275,below left] {$f(x)$};
 end{axis}
end{tikzpicture}
end{document}