[discia] GCSE Maths › Algebra › Inverse and Composite Functions

Inverse and Composite Functions

discia.org 10 questions · 37 marks
Name Date
  1. Either paper 4 marks

    \(f(x) = 2x - 3\) and \(g(x) = 4x - 7\)

    (a)

    Find the value of \(fg(2)\).

    (b)

    Find the value of \(gf(2)\).

    Show answer
    (a) \(-1\)
    (b) \(-3\)
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  2. Either paper 3 marks

    \(f(x) = ax - 5\)

    \(f(3) = 13\)

    Work out the value of \(a\).

    Show answer
    \(6\)
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  3. Either paper 3 marks

    \(f(x) = 7x + 5\) and \(g(x) = 6x - 5\)

    Find \(fg(x)\).

    Give your answer in the form \(px + q\), where \(p\) and \(q\) are integers.

    Show answer
    \(42x - 30\)
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  4. Either paper 3 marks

    \(f(x) = 6x - 6\)

    Find an expression for \(f^{-1}(x)\).

    Show answer
    \(f^{-1}(x) = \frac{x + 6}{ 6 }\)
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  5. Either paper 3 marks

    \(f(x) = 7x - 6\) and \(g(x) = x + 4\)

    \(fg(k) = 50\)

    Find the value of \(k\).

    Show answer
    \(4\)
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  6. Either paper 4 marks

    \(f(x) = 5x + 6\)

    (a)

    Work out \(f^{-1}(x)\).

    (b)

    Solve the equation \(f^{-1}(x) = -3\).

    Show answer
    (a) \(f^{-1}(x) = \frac{x - 6}{ 5 }\)
    (b) \(-9\)
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  7. Either paper 5 marks

    \(f(x) = x^2 - 1\) and \(g(x) = x - 3\)

    (a)

    Work out \(gf(x)\).

    (b)

    Work out \(fg(x)\).

    Give your answer in the form \(x^2 + px + q\).

    Show answer
    (a) \(x^2 - 4\)
    (b) \(x^2 - 6x + 8\)
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  8. Either paper 4 marks

    The graph of \(y = h(x)\) is drawn on the grid.

    xy-7-6-5-4-3-21234567-7-6-5-4-3-212345670
    (a)

    On the grid, draw the graph of the inverse function \(y = h^{-1}(x)\) for \(1 \le x \le 3\). Plot its two end points, then join them.

    (b)

    \(f(x) = \sin x\) and \(g(x) = x + 90\)

    On the grid, draw the graph of the composite function \(y = fg(x)\) for \(0 \le x \le 360\). Plot its points at \(x = 0\), \(90\), \(180\), \(270\) and \(360\), then draw the curve.

    xy90180270360-1-0.50.510
    Show answer
    (a) The line segment from \((1, 0)\) to \((3, 1)\), with gradient \(\frac{1}{2}\)
    (b) The sine curve \(y = \sin(x + 90)\) through \((0, 1)\), \((90, 0)\), \((180, -1)\), \((270, 0)\) and \((360, 1)\)
    (a)xy-7-6-5-4-3-21234567-7-6-5-4-3-212345670
    (b)xy90180270360-1-0.50.510
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  9. Either paper 4 marks

    \(f(x) = 5x - 4\)

    Show that \(4f(x) - 100f^{-1}(x)\) simplifies to an integer.

    Show answer
    The \(x\) terms cancel and \(4f(x) - 100f^{-1}(x) = -96\), an integer.
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  10. Either paper 4 marks

    \(f(x) = x^2\) and \(g(x) = -8x - 12\)

    Solve \(f(x) = g(x)\).

    Show answer
    \(x = -6\) or \(x = -2\)
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discia.org Inverse and Composite Functions · sheet 1
Answers Inverse and Composite Functions · sheet 1
  1. (a) \(-1\)
    (b) \(-3\)
  2. \(6\)
  3. \(42x - 30\)
  4. \(f^{-1}(x) = \frac{x + 6}{ 6 }\)
  5. \(4\)
  6. (a) \(f^{-1}(x) = \frac{x - 6}{ 5 }\)
    (b) \(-9\)
  7. (a) \(x^2 - 4\)
    (b) \(x^2 - 6x + 8\)
  8. (a) The line segment from \((1, 0)\) to \((3, 1)\), with gradient \(\frac{1}{2}\)
    (b) The sine curve \(y = \sin(x + 90)\) through \((0, 1)\), \((90, 0)\), \((180, -1)\), \((270, 0)\) and \((360, 1)\)
    (a)xy-7-6-5-4-3-21234567-7-6-5-4-3-212345670
    (b)xy90180270360-1-0.50.510
  9. The \(x\) terms cancel and \(4f(x) - 100f^{-1}(x) = -96\), an integer.
  10. \(x = -6\) or \(x = -2\)
discia.org Inverse and Composite Functions · sheet 1