Simplify: \( f(x) = rac{2(x - 2)(x + 2)}{x - 2} = 2(x + 2) \), for \( x
["# Simplify ( f(x) = \frac{2(x - 2)(x + 2)}{x - 2} = 2(x + 2) ): A Clear Explanation with Domain Considerations", "Simplifying rational functions is a fundamental skill in algebra, and understanding how to simplify expressions like\n[\nf(x) = \frac{2(x - 2)(x + 2)}{x - 2} = 2(x + 2)\n]\nhelps clarify key concepts such as function simplification, domain restrictions, and transformation. This article explores step-by-step how we simplify this function, why the expression reduces so neatly, and crucially, what values of ( x ) are allowed in the domain.", "---", "### Step-by-Step Simplification", "Start with the original expression:\n[\nf(x) = \frac{2(x - 2)(x + 2)}{x - 2}\n]", "At first glance, the numerator contains ( (x - 2) ) multiplied by ( 2(x + 2) ), and the denominator is ( x - 2 ). When ( x <br/>\neq 2 ), ( x - 2 ) is not zero, and the ( x - 2 ) terms in the numerator and denominator cancel out:", "[\nf(x) = 2(x + 2), \quad \ ext{provided } x <br/>\neq 2.\n]", "Thus, the simplified form of the function is:\n[\nf(x) = 2(x + 2)\n]", "But it’s vital to emphasize that this simplification is only valid when ( x <br/>\neq 2 ), due to division by zero in the original expression.", "---", "### Why the Simplification Fails at ( x = 2 )", "Though ( f(x) ) equals ( 2(x + 2) ) for all ( x <br/>\neq 2 ), the original function is undefined at ( x = 2 ) because substituting ( x = 2 ) yields a division by zero:\n[\n\frac{2(2 - 2)(2 + 2)}{2 - 2} = \frac{0}{0}, \quad \ ext{an indeterminate form}.\n]", "This means the function has a removable discontinuity (or a hole) at ( x = 2 ), even though the simplified expression ( 2(x + 2) ) is defined there. The graph appears as a straight line ( y = 2(x + 2) ) with an open circle at ( x = 2 ).", "---", "### Domain of the Simplified Function", "The domain of ( f(x) ) is all real numbers except ( x = 2 ), written as:\n[\n\boxed{ { x \in \mathbb{R} \mid x <br/>\neq 2 } }\n]", "This domain restriction is essential. Even though algebraically the expression becomes ( 2(x + 2) ), the original function is undefined at ( x = 2 ). Thus, when solving equations, writing functions, or graphing, including the restriction ensures accuracy.", "---", "### Practical Implications and Real-World Usage", "Understanding this simplification and domain limitation has real applications:", "- Graphing: Recognizing the hole at ( x = 2 ) prevents misrepresenting the graph as a complete linear line.\n- Problem-Solving: When solving ( f(x) = k ), substituting ( 2(x + 2) = k ) gives ( x = \frac{k}{2} - 2 ), but verify ( x <br/>\neq 2 ). If ( \frac{k}{2} - 2 = 2 ), then ( k = 8 ), corresponding to the hole.\n- Modeling: In applied contexts, undefined points often represent physical or logical boundaries.", "---", "### Summary", "- The rational function ( f(x) = \frac{2(x - 2)(x + 2)}{x - 2} ) simplifies cleanly to ( f(x) = 2(x + 2) ) algebraically.\n- This simplification excludes ( x = 2 ) due to division by zero.\n- The domain is all real numbers except ( x = 2 ).\n- Including domain restrictions ensures precise function behavior and accurate graphing.", "By mastering this simplification and recognizing its domain limitations, students gain clarity on rational functions—key to advancing in algebra and beyond.", "---", "Keywords: simplify rational function, function simplification, domain restrictions, removable discontinuity, ( f(x) = 2(x + 2) ), excluded values, algebraic simplification, solve rational equations."]
