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- https://math.libretexts.org/Courses/Fresno_City_College/Math_3A%3A_College_Algebra_-_Fresno_City_College/04%3A_Polynomial_and_Rational_Functions/4.06%3A_Rational_FunctionsIn the last few sections, we have worked with polynomial functions, which are functions with non-negative integers for exponents. In this section, we explore rational functions, which have variables i...In the last few sections, we have worked with polynomial functions, which are functions with non-negative integers for exponents. In this section, we explore rational functions, which have variables in the denominator.
- https://math.libretexts.org/Courses/Quinebaug_Valley_Community_College/MAT186%3A_Pre-calculus_-_Walsh/03%3A_Polynomial_and_Rational_Functions/3.07%3A_Rational_FunctionsIn the last few sections, we have worked with polynomial functions, which are functions with non-negative integers for exponents. In this section, we explore rational functions, which have variables i...In the last few sections, we have worked with polynomial functions, which are functions with non-negative integers for exponents. In this section, we explore rational functions, which have variables in the denominator.
- https://math.libretexts.org/Courses/Hartnell_College/MATH_25%3A_PreCalculus_(Abramson_OpenStax)/03%3A_Polynomial_and_Rational_Functions/3.07%3A_Rational_FunctionsIn the last few sections, we have worked with polynomial functions, which are functions with non-negative integers for exponents. In this section, we explore rational functions, which have variables i...In the last few sections, we have worked with polynomial functions, which are functions with non-negative integers for exponents. In this section, we explore rational functions, which have variables in the denominator.
- https://math.libretexts.org/Workbench/Book-_Precalculus_I_for_Highline_College_w/Rational_Inequalities_and_Equations_of_Circles/1.03%3A_Polynomial_and_Rational_Functions/1.3.08%3A_Rational_FunctionsIn the last few sections, we have worked with polynomial functions, which are functions with non-negative integers for exponents. In this section, we explore rational functions, which have variables i...In the last few sections, we have worked with polynomial functions, which are functions with non-negative integers for exponents. In this section, we explore rational functions, which have variables in the denominator.
- https://math.libretexts.org/Courses/Highline_College/Math_141%3A_Precalculus_I_(old_edition)/03%3A_Polynomial_and_Rational_Functions/3.07%3A_Rational_FunctionsIn the last few sections, we have worked with polynomial functions, which are functions with non-negative integers for exponents. In this section, we explore rational functions, which have variables i...In the last few sections, we have worked with polynomial functions, which are functions with non-negative integers for exponents. In this section, we explore rational functions, which have variables in the denominator.
- https://math.libretexts.org/Courses/Mission_College/Math_001%3A_College_Algebra_(Kravets)/05%3A_Polynomial_and_Rational_Functions/5.06%3A_Rational_FunctionsIn the last few sections, we have worked with polynomial functions, which are functions with non-negative integers for exponents. In this section, we explore rational functions, which have variables i...In the last few sections, we have worked with polynomial functions, which are functions with non-negative integers for exponents. In this section, we explore rational functions, which have variables in the denominator.
- https://math.libretexts.org/Courses/Coastline_College/Math_C170%3A_Precalculus_(Tran)/03%3A_Polynomial_and_Rational_Functions/3.08%3A_Rational_FunctionsIn the last few sections, we have worked with polynomial functions, which are functions with non-negative integers for exponents. In this section, we explore rational functions, which have variables i...In the last few sections, we have worked with polynomial functions, which are functions with non-negative integers for exponents. In this section, we explore rational functions, which have variables in the denominator.
- https://math.libretexts.org/Courses/Cosumnes_River_College/Math_372%3A_College_Algebra_(Lecture_Notes)/02%3A_Polynomial_and_Rational_Functions/2.07%3A_Rational_FunctionsIf a rational function has \( x \)-intercepts at \(x= x_1 , x_2 , \ldots , x_n \), vertical asymptotes at \(x= v_1 , v_2 , \ldots , v_m \), and \( x i \neq v_j \) for all \( i \) and \( j \), then the...If a rational function has \( x \)-intercepts at \(x= x_1 , x_2 , \ldots , x_n \), vertical asymptotes at \(x= v_1 , v_2 , \ldots , v_m \), and \( x i \neq v_j \) for all \( i \) and \( j \), then the function can be written in the form:\[f(x) = a \dfrac{(x- x_1 )^{p_1} (x- x_2 )^{p_2} \cdots (x- x_n )^{p_n}}{(x- v_1 )^{q_1} (x- v_2 )^{q_2} \cdots (x- v_m )^{q_m}}, \nonumber \]where the powers \( p_i \) or \( q_i \) on each factor can be determined by the behavior of the graph at the correspond…
- https://math.libretexts.org/Bookshelves/Precalculus/Precalculus_2e_(OpenStax)/03%3A_Polynomial_and_Rational_Functions/3.08%3A_Rational_FunctionsIn the last few sections, we have worked with polynomial functions, which are functions with non-negative integers for exponents. In this section, we explore rational functions, which have variables i...In the last few sections, we have worked with polynomial functions, which are functions with non-negative integers for exponents. In this section, we explore rational functions, which have variables in the denominator.
- https://math.libretexts.org/Courses/Cosumnes_River_College/Math_372%3A_College_Algebra_for_Calculus_(2e)/02%3A_Polynomial_and_Rational_Functions/2.07%3A_Rational_FunctionsThis section introduces rational functions, exploring their key features such as domain, vertical and horizontal or slant asymptotes, and intercepts. It discusses how to analyze and graph rational fun...This section introduces rational functions, exploring their key features such as domain, vertical and horizontal or slant asymptotes, and intercepts. It discusses how to analyze and graph rational functions, focusing on behavior near asymptotes and at infinity. Examples demonstrate the steps to identify these features and interpret their implications for the function's graph.
