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Functions Toolkit (DP IB Applications & Interpretation (AI))

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    What is a composite function?

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  • What is a composite function?

    A composite function is a function made up of two or more functions. One function is applied to another function, e.g. open parentheses f ring operator g close parentheses open parentheses x close parentheses.

  • True or False?

    The order in which the functions in a composite function are applied does not matter.

    False.

    The order in which the functions in a composite function are applied does matter. open parentheses f ring operator g close parentheses open parentheses x close parentheses is not usually equivalent to open parentheses g ring operator f close parentheses open parentheses x close parentheses.

    Always apply the function closest to the variable first.

    E.g. for the composite function open parentheses f ring operator g close parentheses open parentheses x close parentheses, you must first apply the function g followed by the function f.

  • To find the domain and range of a composite function, what is the first step you need to take?

    To find the domain and range of a composite function, the first step is you need to take is to find the range of the first function that is applied.

    E.g. To find the domain and range for the function open parentheses f ring operator g close parentheses open parentheses x close parentheses, you must first find the range of g.

  • True or False?

    An inverse function reverses the effect of a function.

    True.

    An inverse function reverses the effect of a function.

  • True or False?

    Only a many-to-one function has an inverse function.

    False.

    A many-to-one function does not have an inverse function.

    Only one-to-one functions have inverses.

  • What is the horizontal line test?

    The horizontal line test is a method to determine if a function has an inverse.

    If a function has an inverse, then any horizontal line should intersect with the graph once at most.

  • What is the relationship between the domain of a function and its inverse function?

    The domain of a function becomes the range of its inverse function.

  • State the relationship between the graphs of f open parentheses x close parenthesesand f to the power of negative 1 end exponent open parentheses x close parentheses.

    The graph y equals f to the power of negative 1 end exponent open parentheses x close parentheses is a reflection of the graph y equals f open parentheses x close parentheses in the line y equals x.

    A set of axes showing the graph y = f(x) and it's inverse y = f^-1(x) as a reflection in the line y = x.

  • What is the identity function?

    The identity function maps each value in a function to itself.

    id open parentheses x close parentheses equals x.

  • If two composite functions,space f ring operator g and space g ring operator f, have the same effect as the identity function, what can be said about the functions space f and space g? 

    If two composite functions,space f ring operator g and space g ring operator f, have the same effect as the identity function, then the functions space f and space g must be inverse functions of each other.

  • How can you adapt a many-to-one function in order for it to have an inverse?

    You can restrict the domain of a many-to-one function to a subset of the domain on which the function is one-to-one. Then the function on the restricted domain will have an inverse.

    E.g. the function y equals x squared is a many-to-one function, but if the domain is restricted to x greater or equal than 0, the function becomes one-to-one and has an inverse.

  • How should the domain for the sine function be restricted in order for it to have an inverse?

    In order for the sine function to have an inverse, the domain must be restricted to half a cycle, from a maximum to a minimum or vice versa.

    E.g. negative pi over 2 less or equal than x less or equal than pi over 2.

  • True or False?

    For a many-to-one function with a restricted domain, the range of its inverse function is the same as the original function's domain.

    False.

    For a many-to-one function with a restricted domain, the range of its inverse function is the not the same as the original function's domain.

    The range of the inverse function is the same as the restricted domain of the original function.

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    1. Topic 1

      Number Toolkit

    2. Topic 2

      Exponentials & Logs

    3. Topic 3

      Sequences & Series

    4. Topic 4

      Financial Applications

    5. Topic 5

      Complex Numbers

    6. Topic 6

      Further Complex Numbers

    7. Topic 7

      Matrices

    8. Topic 8

      Eigenvalues & Eigenvectors

    1. Topic 1

      Linear Functions & Graphs

    2. Topic 2

      Further Functions & Graphs

    3. Topic 3

      Modelling with Functions

    4. Topic 4

      Functions Toolkit

    5. Topic 5

      Transformations of Graphs

    6. Topic 6

      Modelling with Logarithmic, Logistic & Piecewise Functions

    1. Topic 1

      Geometry Toolkit

    2. Topic 2

      Geometry of 3D Shapes

    3. Topic 3

      Trigonometry

    4. Topic 4

      Trigonometric Identities & Equations

    5. Topic 5

      Voronoi Diagrams

    6. Topic 6

      Matrix Transformations

    7. Topic 7

      Vector Properties

    8. Topic 8

      Vector Equations of Lines

    9. Topic 9

      Modelling with Vectors

    10. Topic 10

      Graph Theory

    1. Topic 1

      Statistics Toolkit

    2. Topic 2

      Correlation & Regression

    3. Topic 3

      Non-linear Regression

    4. Topic 4

      Probability

    5. Topic 5

      Probability Distributions

    6. Topic 6

      Random Variables

    7. Topic 7

      Binomial Distribution

    8. Topic 8

      Normal Distribution

    9. Topic 9

      Combinations of Normal Distributions & Sample Mean Distributions

    10. Topic 10

      Poisson Distribution

    11. Topic 11

      Hypothesis Testing using the Chi-squared Distribution

    12. Topic 12

      Hypothesis Testing for Population Parameters

    13. Topic 13

      Transition Matrices & Markov Chains

    1. Topic 1

      Differentiation

    2. Topic 2

      Further Differentiation

    3. Topic 3

      Integration

    4. Topic 4

      Further Integration

    5. Topic 5

      Kinematics

    6. Topic 6

      Differential Equations

    7. Topic 7

      Coupled & Second Order Differential Equations