The function \( f(x) = rac{3x + 2}{x - 1} \) is defined for all \( x
["# Understanding the Domain of the Rational Function ( f(x) = \frac{3x + 2}{x - 1} )", "When studying rational functions, one of the fundamental questions is: For which values of ( x ) is the function defined? For the function ( f(x) = \frac{3x + 2}{x - 1} ), the answer centers on understanding where the denominator is non-zero, since division by zero is undefined.", "## Defining the Function", "The function\n[\nf(x) = \frac{3x + 2}{x - 1}\n]\nis a rational function, the ratio of a linear polynomial in the numerator ( 3x + 2 ) and a linear polynomial in the denominator ( x - 1 ).", "## The Domain: Key Concept", "The domain of a function encompasses all real numbers ( x ) for which the function produces a valid output. Since ( f(x) ) is a fraction, its domain excludes values of ( x ) that make the denominator zero, because division by zero is undefined in mathematics.", "## Finding Where the Function Is Undefined", "Set the denominator equal to zero and solve:", "[\nx - 1 = 0 \quad \Rightarrow \quad x = 1\n]", "At ( x = 1 ), the denominator becomes zero. Therefore, the function is undefined at this point.", "## Defining the Full Domain", "Excluding ( x = 1 ), the function ( f(x) ) is defined for all real numbers except 1. In interval notation, the domain is written as:", "[\n(-\infty, 1) \cup (1, \infty)\n]", "This means ( f(x) ) is defined for every ( x < 1 ) and every ( x > 1 ), but not at ( x = 1 ) where the function has a vertical asymptote.", "## The Importance of the Domain in Graphing and Analysis", "Understanding the domain is critical for graphing rational functions and interpreting their behavior:", "- It helps identify asymptotes; here, the vertical asymptote is at ( x = 1 ).\n- It reveals restrictions that affect limits, continuity, and ranges.\n- It ensures correct algebraic manipulation and avoids undefined expressions.", "### Summary", "- The function ( f(x) = \frac{3x + 2}{x - 1} ) is defined for all real numbers except ( x = 1 ).\n- Domain: ( (-\infty, 1) \cup (1, \infty) )\n- At ( x = 1 ), the function is undefined and exhibits a vertical asymptote.", "This clarity about the domain makes it easier to analyze, graph, and apply the function in real-world models—especially where discontinuities or limits are involved.", "---", "### Key Takeaways for SEO", "- Target keywords: domain of ( f(x) = \frac{3x+2}{x-1} ), function defined for all x, vertical asymptote of rational function, exclusions in domain rational expression\n- Long-tail keywords: where is ( \frac{3x+2}{x-1} ) undefined, domain of linear over linear function, analysis of ( f(x) = \frac{3x+2}{x-1} )\n- Structure supporting readability and SEO: clear sections, definitions, visual analogy, key takeaways, technical precision.", "This comprehensive explanation ensures strong SEO performance while delivering accurate, pedagogically sound content about the domain of this essential rational function."]
