Subjects
Grades
The Rational Function
Concept sheet | Mathematics
Secondary
5
Rational functions include a fraction with an independent variable |(x)| in the denominator.
The rational function is often associated with the inverse variation function (inversely proportional situation). In reality, the inverse variation function is a special set of rational functions.
The rule of an inverse variation function is |f(x)=\dfrac{k}{x}| where |x \neq 0.|
Related Concepts
- The Role of the Parameters in a Rational Function in Standard Form
- Switching from the Standard to the General Form of a Rational Function
- Graphing a Rational Function
- Finding the Rule of a Rational Function
- The Properties of the Rational Function
- Solving a Rational Equation or Inequality
- Solving Problems Involving Rational Functions
- The Inverse of the Rational Function
The Basic Rational Function
The basic form of a rational function is the following.
||f(x)=\dfrac{1}{x}|| where |x \neq 0|
The Graph and Properties of the Basic Rational Function
The graph of the basic rational function is composed of 2 curves, called hyperbolas, which approach the 2 axes without touching them.
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One curve intersects the point |(1,1),| and the other the point |(-1,-1).|
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The domain is |\mathbb{R}\backslash \{0\}.|
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The range is |\mathbb{R}\backslash \{0\}.|
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The function is decreasing over the interval |]-\infty,0[,| and then over the interval |]0,+\infty[.|
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The basic function has no |y|-intercept or zero.
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The function has no maximum or minimum.
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The function has 2 perpendicular asymptotes: |x=0| and |y=0.|
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The function is negative over the interval |]-\infty,0[| and is positive over the interval |]0,+\infty[.|
The Transformed Rational Function
The transformed rational function can be written in 2 different ways.
The standard form
||f(x)=\dfrac{a}{b(x-h)}+k|| where |b\neq 0|
The general form
||f(x)=\dfrac{a_1 x+b_1}{a_2 x+b_2}|| where |a_2 \neq 0|
The Graph and Properties of the Transformed Rational Function
The graph of the transformed rational function is composed of 2 curves, called hyperbolas, which approach the 2 asymptotes without touching them.
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The domain is |\mathbb{R}\backslash \{h\}.|
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The range is |\mathbb{R}\backslash \{k\}.|
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When |ab>0,| the function is decreasing over the interval |]-\infty,h[,| and then over the interval |]h,+\infty[.|
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When |ab<0,| the function is increasing over the interval |]-\infty,h[,| and then over the interval |]h,+\infty[.|
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The function has a |y|-intercept if |h\neq 0| and a zero if |k \neq 0.|
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The function has no maximum or minimum.
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The function has 2 perpendicular asymptotes: |x=h| and |y=k.|
