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TMUA 2023 · Paper 1 · Question 20 of 20

TMUA 2023 Paper 1 Question 20

Graphs and transformations — Composite of a quadratic with a bounded function. Try it first; the answer and a full worked solution are below.

TMUA 2023 · Paper 1Graphs and transformationsComposite of a quadratic with a bounded function8 optionshard

The original question includes a diagram: The graph of y = f(x), rising to a maximum of 2 at x = 1 and falling to a minimum of -2 at x = -1, tending to zero far from the origin.

The diagram shows the graph of y = f(x). The function f attains its maximum value of 2 at x = 1, and its minimum value of −2 at x = −1.

Find the difference between the maximum and minimum values of (f(x))2 − f(x).

  1. A2
  2. B94
  3. C4
  4. D174
  5. E6
  6. F254
  7. G8
  8. H334
Show the answer and worked solution
answer · F
  1. A2
  2. B94
  3. C4
  4. D174
  5. E6
  6. F254
  7. G8
  8. H334
Put u = f(x). The graph is continuous with maximum 2 and minimum −2, so u takes every value in [−2,  2], and the question becomes: what is the range of g(u) = u2 − u on that interval? Its vertex is at u = 12, where g = −14, and the endpoints give g(−2) = 6 and g(2) = 2. So the greatest value is 6 and the least is −14, a difference of 254.