Reciprocal & Rational Functions (DP IB Analysis & Approaches (AA)): Revision Note

Reciprocal functions & graphs

What is the reciprocal function?

• The reciprocal function is defined by

• Its domain is the set of all real values except 0

• Its range is the set of all real values except 0

• The reciprocal function has a self-inverse nature

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What are the key features of the reciprocal graph?

• The graph does not have a y-intercept

• The graph does not have any roots

• The graph has two asymptotes

  • A horizontal asymptote at the x-axis: 

    • This is the limiting value when the absolute value of x gets very large

  • A vertical asymptote at the y-axis: 

    • This is the value that causes the denominator to be zero

• The graph has two axes of symmetry

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• The graph does not have any minimum or maximum points

Linear rational functions & graphs

What is a rational function with linear expressions?

• A (linear) rational function is of the form 

  • e.g.

• The reciprocal function is a special case of a rational function

• The inverse is also a rational function

  • You do not need to remember this formula

    • You can derive the inverse easily in your exam

What are the key features of linear rational graphs?

Intersections with coordinate axes

• The graph has a y-intercept at provided 

  • Substitute to find the y-coordinate

    • e.g. the y-intercept of is

    • e.g. does not have a y-intercept

• The graph has one root at provided 

  • Set the numerator equal to zero and solve

    • e.g. the root of is

    • e.g. does not have any roots

Asymptotes

• The graph has two asymptotes

  • A horizontal asymptote: 

    • This is the limiting value when the absolute value of x gets very large

    • e.g. the horizontal asymptote of is

  • A vertical asymptote: 

    • This is the value that causes the denominator to be zero

    • e.g. the horizontal asymptote of is

Domain and range

• Its domain is the set of all real values except 

  • It is undefined for the value of which causes the denominator to equal zero

    • e.g. the domain of is

• Its range is the set of all real values except 

  • e.g. the range of is

Turning points

• The graph does not have any minimum or maximum points