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Composite and Inverse Functions Calculator

Work out composite and inverse functions the way GCSE Higher sets them. Give f as a linear function and g as a quadratic or linear one, then choose fg(x), gf(x), ff(x), the inverse f⁻¹(x), or a single value such as fg(2). Every substitution is written out, inside function first, so the order never gets muddled.

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Method, Step by Step

    Worked Example

    f(x) = 2x + 3 and g(x) = x² − 1. Find fg(x).

    1. fg(x) = f(g(x)), so g goes in first
    2. Replace x in f with g(x):fg(x) = 2(x² − 1) + 3
    3. Multiply out:2x² − 2 + 3 = 2x² + 1

    Answer: fg(x) = 2x² + 1

    Try These Yourself

    f(x) = 3x − 4. Find f⁻¹(x).Show answerHide answer

    Answer: f⁻¹(x) = (x + 4) ÷ 3

    y = 3x − 4, so y + 4 = 3x and x = (y + 4) ÷ 3.

    f(x) = 2x + 3 and g(x) = x² − 1. Find gf(1).Show answerHide answer

    Answer: 24

    f(1) = 5, then g(5) = 25 − 1 = 24.

    f(x) = x + 5. Find ff(x).Show answerHide answer

    Answer: ff(x) = x + 10

    f(x + 5) = (x + 5) + 5.