20. (Total Score: 14 points) Subquestion 1 is worth 6 points, Subquestion 2 is worth 8 points A quadratic function $y = f _ { 1 } ( x )$ has its vertex at the origin and passes through the point $(1,1)$. A reciprocal function $y = f _ { 2 } ( x )$ has its graph intersecting the line $y = x$ at two points with distance 8 between them. Let $f ( x ) = f _ { 1 } ( x ) + f _ { 2 } ( x )$. (1) Find the expression for function $f ( x )$; (2) Prove: When $a > 3$, the equation $f ( x ) = f ( a )$ in $x$ has three real solutions.
20. (Total Score: 14 points) Subquestion 1 is worth 6 points, Subquestion 2 is worth 8 points\\
A quadratic function $y = f _ { 1 } ( x )$ has its vertex at the origin and passes through the point $(1,1)$. A reciprocal function $y = f _ { 2 } ( x )$ has its graph intersecting the line $y = x$ at two points with distance 8 between them. Let $f ( x ) = f _ { 1 } ( x ) + f _ { 2 } ( x )$.\\
(1) Find the expression for function $f ( x )$;\\
(2) Prove: When $a > 3$, the equation $f ( x ) = f ( a )$ in $x$ has three real solutions.