For which pair of functions is $$ (f \circ g)(x)=x $$ ? $$ f(x)=\frac{x^{2}}{x} $$ and $$ g(x)=\frac{1}{x} $$ $$ f(x)=\frac{x-2}{3} $$ and $$ g(x)=\frac{2}{x} $$ $$ f(x)=\frac{1}{2} x-2 $$ and $$ g(x)=\frac{1}{2} x+2 $$

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1. For the first pair, we have 
   $$
   f(x)=\frac{x^2}{x}.
   $$
   For $$x\neq0$$ this simplifies to 
   $$
   f(x)=x.
   $$
   The second function is 
   $$
   g(x)=\frac{1}{x}.
   $$
   Composing, we get 
   $$
   (f\circ g)(x)=f(g(x))=f\left(\frac{1}{x}\right)=\frac{1}{x}.
   $$
   Clearly, $$\frac{1}{x}\ne x$$ in general. It equals $$x$$ only when
   $$
   \frac{1}{x}=x \quad\Longrightarrow\quad x^2=1,
   $$
   i.e. for $$x=1$$ or $$x=-1$$ only. Thus, this pair does not produce $$(f\circ g)(x)=x$$ for all $$x$$.

2. For the second pair, we have 
   $$
   f(x)=\frac{x-2}{3} \quad \text{and} \quad g(x)=\frac{2}{x}.
   $$
   Then,
   $$
   (f\circ g)(x)=f\left(\frac{2}{x}\right)=\frac{\frac{2}{x}-2}{3}=\frac{2-2x}{3x}.
   $$
   To require $$(f\circ g)(x)=x$$ for all $$x$$, we would need
   $$
   \frac{2-2x}{3x}=x.
   $$
   Multiplying both sides by $$3x$$ (assuming $$x\neq 0$$) gives:
   $$
   2-2x=3x^2.
   $$
   This is a quadratic equation in $$x$$ and is not an identity; it holds only for particular $$x$$. Thus, this pair does not satisfy $$(f\circ g)(x)=x$$ in general.

3. For the third pair, 
   $$
   f(x)=\frac{1}{2}x-2 \quad \text{and} \quad g(x)=\frac{1}{2}x+2.
   $$
   The composition is
   $$
   (f\circ g)(x)=f\left(\frac{1}{2}x+2\right)=\frac{1}{2}\left(\frac{1}{2}x+2\right)-2.
   $$
   This simplifies to:
   $$
   (f\circ g)(x)=\frac{1}{4}x+1-2=\frac{1}{4}x-1.
   $$
   For $$(f\circ g)(x)$$ to be equal to $$x$$ for all $$x$$, we would require:
   $$
   \frac{1}{4}x-1=x.
   $$
   Rearranging, 
   $$
   -1=x-\frac{1}{4}x=\frac{3}{4}x \quad\Longrightarrow\quad x=-\frac{4}{3}.
   $$
   This equality holds only at $$x=-\frac{4}{3}$$ and not for all $$x$$.

None of the three pairs satisfy $$(f\circ g)(x)=x$$ for all values of $$x$$.
None of the given pairs of functions satisfy $$(f \circ g)(x) = x$$ for all $$x$$.

For which pair of functions is ?
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