\[ f(x) = rac{(x-1)(x+1)}{x-1} \]

["# Understanding ( f(x) = \frac{(x-1)(x+1)}{x-1} ): Simplifying the Function", "When encountering the function\n[ f(x) = \frac{(x-1)(x+1)}{x-1}, ]\nit’s essential to recognize both its algebraic structure and behavior. This article explores the function step by step, including simplification, domain considerations, key features, and applications — all optimized for search engines to help students, educators, and math enthusiasts fully grasp this important rational expression.", "---", "## What Is ( f(x) = \frac{(x-1)(x+1)}{x-1} )?", "This function is a rational expression involving a fraction where both the numerator and denominator contain a common factor: ( (x - 1) ). At first glance, simplifying the expression seems straightforward, but understanding its implications requires careful analysis.", "---", "## Simplifying the Function", "We start with:\n[\nf(x) = \frac{(x-1)(x+1)}{x-1}\n]", "### For ( x <br/>\ne 1 ),\nthe common factor ( (x - 1) ) in the numerator and denominator cancels:\n[\nf(x) = x + 1, \quad \ ext{for } x <br/>\ne 1\n]", "However, note that cancellation is valid only when ( x - 1 <br/>\ne 0 ), i.e., ( x <br/>\ne 1 ). At ( x = 1 ), the original function is undefined because division by zero occurs.", "Thus, the simplified expression is:\n[\nf(x) = x + 1, \quad \ ext{with a hole at } x = 1\n]", "---", "## Domain of the Function", "Since division by zero is undefined, the domain excludes ( x = 1 ). Therefore:\nDomain:\n[\nD = { x \in \mathbb{R} \mid x <br/>\ne 1 }\n]", "Graphically, this means a straight line ( y = x + 1 ) with a removable discontinuity (a "hole") labeled at ( x = 1 ).", "---", "## Behavior Near the Discontinuity", "At ( x = 1 ), though the function is undefined, we can examine the limit:", "[\n\lim_{x \ o 1} f(x) = \lim_{x \ o 1} (x + 1) = 2\n]", "This reveals a removable discontinuity or hole at the point ( (1, 2) ). While the function value doesn’t exist there, nearby values approach 2.", "---", "## Graph of the Function", "- The simplified form ( y = x + 1 ) is a straight line with slope 1 and y-intercept 1.\n- At ( x = 1 ), there is a hole, not an actual point on the graph.\n- Vertically, there’s a vertical asymptote or Hole — since the factor cancels, there is no asymptote, but the discontinuity marks the undefined point clearly.", "---", "## Key Features of ( f(x) )", "| Feature | Description |\n|---------------------|---------------------------------------------------------|\n| Simplified Form | ( f(x) = x + 1 ), for ( x <br/>\ne 1 ) |\n| Domains | ( x <br/>\ne 1 ) |\n| Discontinuity | Removable hole at ( x = 1 ), ( f(1) ) undefined |\n| Range | All real numbers except ( f(1) = 2 ) |\n| Slope & Intercept | Line slope = 1, y-intercept = 1, but shifted down by 1 at ( x=1 ) |", "---", "## Practical Applications and Algebraic Significance", "Understanding such functions is crucial in algebra, calculus, and real-world modeling. For instance:", "- Modeling relationships with undefined points, such as cost functions where inputs cannot be negative or zero.\n- Teaching removable discontinuities — a common concept before learning asymptotes and more complex rational functions.\n- Serving as a mental model for simplifying expressions with common factors.", "---", "## Why Avoid Divide-by-Zero?", "This example highlights a fundamental rule: canceling factors only when they’re not zero. Cancelling ( (x - 1) ) assumes ( x <br/>\ne 1 ). Ignoring this leads to confusion about undefined points — using this clarifies function behavior and limits.", "---", "## Conclusion and Further Reading", "The function ( f(x) = \frac{(x-1)(x+1)}{x-1} ) appears simple but offers depth in understanding algebraic simplification, domains, and removable discontinuities. For students and educators, mastering such examples builds a strong foundation for more advanced calculus and algebra topics.", "Further Explore:", "- How do removable discontinuities differ from vertical asymptotes?\n- How to factor and simplify other rational expressions.\n- Limits involving hole functions.", "---", "### Key SEO Keywords:\n- Simplify ( f(x) = \frac{(x-1)(x+1)}{x-1} )\n- Removable discontinuity in rational functions\n- Simplify algebraic expressions with common factors\n- Domain of ( f(x) = \frac{(x-1)(x+1)}{x-1} )\n- Understanding holes in graphs\n- How to simplify rational expressions", "---", "Incorporating clear structure, thorough explanation, and targeted keywords ensures this SEO-friendly article ranks well and serves readers seeking mastery over this foundational calculus concept."]