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IBMaths: AA SLDPRevision Notes2. Functions2.4 Further Functions & Graphs2.4.1 Reciprocal & Rational Functions

Reciprocal & Rational Functions (DP IB Maths: AA SL)

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Dan

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Dan

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Reciprocal Functions & Graphs

What is the reciprocal function?

  • The reciprocal function is defined by space f left parenthesis x right parenthesis equals 1 over x comma space x not equal to 0
  • 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
    • space f to the power of negative 1 end exponent left parenthesis x right parenthesis equals f left parenthesis x right parenthesis
    • left parenthesis f ring operator f right parenthesis left parenthesis x right parenthesis equals x

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: space y equals 0
      • This is the limiting value when the absolute value of x gets very large
    • A vertical asymptote at the y-axis: space x equals 0
      • This is the value that causes the denominator to be zero
  • The graph has two axes of symmetry
    • y equals x
    • y equals negative x
  • The graph does not have any minimum or maximum points

2-4-1-ib-aa-sl-reciprocal-function

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Linear Rational Functions & Graphs

What is a rational function?

  • A rational function is of the form space f left parenthesis x right parenthesis equals fraction numerator a x plus b over denominator c x plus d end fraction comma space x not equal to negative d over c
  • Its domain is the set of all real values except  negative d over c
  • Its range is the set of all real values except a over c
  • The reciprocal function is a special case of a rational function

What are the key features of rational graphs?

  • The graph has a y-intercept at stretchy left parenthesis 0 comma space b over d stretchy right parenthesis provided d not equal to 0
  • The graph has one root at stretchy left parenthesis negative b over a comma space 0 stretchy right parenthesis provided a not equal to 0
  • The graph has two asymptotes
    • A horizontal asymptote: space y equals a over c
      • This is the limiting value when the absolute value of x gets very large
    • A vertical asymptote: space x equals negative d over c
      • This is the value that causes the denominator to be zero
  • The graph does not have any minimum or maximum points
  • If you are asked to sketch or draw a rational graph:
    • Give the coordinates of any intercepts with the axes
    • Give the equations of the asymptotes

2-4-1-ib-aa-sl-rational-function

Examiner Tip

  • If you draw a horizontal line anywhere it should only intersect this type of graph once at most
  • The only horizontal line that should not intersect the graph is the horizontal asymptote
    • This can be used to check your sketch in an exam

Worked example

The function space f is defined by space f left parenthesis x right parenthesis equals fraction numerator 10 minus 5 x over denominator x plus 2 end fraction for x not equal to negative 2.

a)
Write down the equation of
(i)
the vertical asymptote of the graph of space f,
(ii)
the horizontal asymptote of the graph of space f.

2-4-1-ib-aa-sl-rational-func-a-we-solution

b)
Find the coordinates of the intercepts of the graph of space f with the axes.

2-4-1-ib-aa-sl-rational-func-b-we-solution

c)
Sketch the graph of space f.

2-4-1-ib-aa-sl-rational-func-c-we-solution

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    Previous:2.3.3 Graphing FunctionsNext:2.4.2 Exponential & Logarithmic Functions
    Dan

    Author: Dan

    Expertise: Maths

    Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.