The function \( R(x) = rac{x^2 - 4x + 3}{x - 1} \) is undefined where the denominator is zero. Set the denominator equal to zero:

["Understanding Where the Function ( R(x) = \frac{x^2 - 4x + 3}{x - 1} ) Is Undefined", "The function ( R(x) = \frac{x^2 - 4x + 3}{x - 1} ) is a rational function, meaning it is defined as a fraction with a polynomial in the numerator and a polynomial in the denominator. However, like all rational functions, ( R(x) ) is undefined whenever the denominator equals zero—this occurs at values of ( x ) that make the denominator equal to zero.", "### Why is the Function Undefined Where Denominator is Zero?", "Mathematically, division by zero is undefined in real numbers. When solving for where ( R(x) ) is undefined, we set the denominator equal to zero:", "[\nx - 1 = 0\n]", "Solving for ( x ), we find:", "[\nx = 1\n]", "Thus, the function ( R(x) ) is undefined at ( x = 1 ), because at this point the denominator becomes zero, violating the fundamental rules of division.", "### Behavior Near ( x = 1 )", "Even though ( x = 1 ) is not in the domain of ( R(x) ), analyzing the numerator helps clarify what happens near this point. Factoring the numerator:", "[\nx^2 - 4x + 3 = (x - 1)(x - 3)\n]", "So the function can be rewritten as:", "[\nR(x) = \frac{(x - 1)(x - 3)}{x - 1}\n]", "For all ( x <br/>\neq 1 ), the ( x - 1 ) terms cancel, simplifying to:", "[\nR(x) = x - 3\n]", "However, as a rational function, we strictly consider the original expression. At ( x = 1 ), the function remains undefined because the simplified form ( x - 3 ) might suggest a removable discontinuity (a "hole"), but the original function still has a division by zero at this point.", "### Conclusion", "The rational function ( R(x) = \frac{x^2 - 4x + 3}{x - 1} ) is undefined where the denominator is zero — specifically at ( x = 1 ). While simplifying the expression reveals simplification potential, the undefined point must still be acknowledged in the original form. Understanding where rational functions are undefined ensures accurate graphing, equation solving, and analysis in algebra and calculus.", "---", "Key Term: Set denominator equal to zero: ( x - 1 = 0 \Rightarrow x = 1 ). This is where ( R(x) ) is undefined."]