Functions

    OCR
    GCSE

    Functions are defined as mappings between a domain (input) and a range (output), strictly requiring that every element in the domain maps to exactly one element in the range. Mastery involves manipulating function notation, constructing composite functions where the output of one function becomes the input of another, and deriving inverse functions through algebraic rearrangement. Analysis extends to graphical interpretations, specifically the reflection of functions in the line y=x to generate inverses, and the rigorous determination of domain restrictions required to ensure functions remain one-to-one.

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    Objectives
    3
    Exam Tips
    4
    Pitfalls
    5
    Key Terms
    4
    Mark Points

    What You Need to Demonstrate

    Key skills and knowledge for this topic

    • Award M1 for correct substitution of the inner function into the outer function (e.g., replacing every x in f with the expression for g)
    • Award M1 for the specific method step of swapping x and y (or equivalent) when determining an inverse function
    • Award A1 for a fully simplified algebraic expression; partial simplification often results in lost accuracy marks
    • Award B1 for correct evaluation of a function at a specific numerical value, observing strict order of operations

    Example Examiner Feedback

    Real feedback patterns examiners use when marking

    • "You calculated f(x) * g(x) here. Remember, fg(x) means you must put the entire expression for g into f."
    • "Good start finding the inverse, but you forgot to swap x and y before rearranging."
    • "Your expansion of (x-2)^2 was incorrect. Be careful with the middle term when squaring binomials inside functions."
    • "You found the expression for the function correctly, but the question asked you to 'Solve' f(x) = 0, so you need to find the value of x."

    Marking Points

    Key points examiners look for in your answers

    • Award M1 for correct substitution of the inner function into the outer function (e.g., replacing every x in f with the expression for g)
    • Award M1 for the specific method step of swapping x and y (or equivalent) when determining an inverse function
    • Award A1 for a fully simplified algebraic expression; partial simplification often results in lost accuracy marks
    • Award B1 for correct evaluation of a function at a specific numerical value, observing strict order of operations

    Examiner Tips

    Expert advice for maximising your marks

    • 💡When calculating fg(x), always write the expression for g(x) inside the brackets of f first (e.g., f[g(x)]) before expanding; this reduces algebraic errors.
    • 💡For inverse functions, explicitly write 'let y = ...' and show the rearrangement step clearly; method marks are often available even if the final algebra goes wrong.
    • 💡Pay close attention to the command word 'Solve' versus 'Find'; 'Solve' implies finding a specific value for x, usually resulting from setting two functions equal to each other.

    Common Mistakes

    Pitfalls to avoid in your exam answers

    • Interpreting the composite function fg(x) as the product f(x) * g(x) rather than applying f to the output of g
    • Confusing the inverse function notation f^-1(x) with the reciprocal 1/f(x)
    • Failing to expand brackets correctly when substituting a negative or complex expression into a quadratic function (e.g., (x-3)^2)
    • Omitting the domain restrictions when they are critical to the definition of the inverse function

    Key Terminology

    Essential terms to know

    Function notation and mapping definitions
    Composite functions and order of operations
    Inverse functions and self-inverse properties
    Domain and range determination
    Graphical representation of inverses

    Likely Command Words

    How questions on this topic are typically asked

    Find
    Solve
    Evaluate
    Express
    Explain

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    Practice questions tailored to this topic