What 9 Among the following four groups of functions, which group contains two functions that are the same?
(1) $f(x)=|x|, g(t)=\sqrt{t^{2}}$;
(2) $f(x)=\sqrt{x^{2}}, g(x)=(\sqrt[3]{x})^{3}$;
(3) $f(x)=\frac{x^{2}-1}{x-1}, g(x)=x+1$;
(4) $f(x)=\sqrt{x+1} \cdot \sqrt{x-1}, g(x)=\sqrt{x^{2}-1}$.
(1) The function $g(t)=\sqrt{t^{2}}$ is equivalent to $g(t)=|t|$, so the function $f(x)=|x|$ and $g(t)=\sqrt{t^{2}}$ have the same correspondence rule and the same domain, making them the same function. (2) The correspondence rule for the function $f(x)=\sqrt{x^{2}}$ is $f(x)=|x|$, while the correspondence rule for the function $g(x)=(\sqrt[3]{x})^{3}$ is $g(x)=x$, and the two have different correspondence rules. (3) The domains of the functions $f(x)=\frac{x^{2}-1}{x-1}$ and $g(x)=x+1$ are $\{x \mid x \neq 1, x \in \mathbf{R}\}$ and $\mathbf{R}$, respectively, and the two have different domains. (4) The domains of the functions $f(x)=\sqrt{x+1} \cdot \sqrt{x-1}$ and $g(x)=\sqrt{x^{2}-1}$ are $\{x \mid x \geqslant 1\}$ and $\{x \mid x \leqslant-1$ or $x \geqslant 1\}$, respectively, and the two have different domains.
Therefore, only the functions in group (1) are the same.
Key points for solving the problem: Two functions are the same only if they have the same correspondence rule and the same domain. Sometimes, the correspondence rule of a function can be transformed through algebraic operations to have the same form as another function's correspondence rule, as in groups (3) and (4) of this problem. However, such operations may change the range of values for the independent variable, so when studying the domain of a function, it is necessary to determine the domain based on the original, most basic correspondence rule given.