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2022 HSC Mathematics Extension 1

Question 9:
Inverse function

1 markinverse functions

A given function has an inverse .

The derivatives of and exist for all real numbers .

The graphs and have at least one point of intersection.

Which statement is true for all points of intersection of these graphs?

  1. All points of intersection lie on the line .
  2. None of the points of intersection lie on the line .
  3. At no point of intersection are the tangents to the graphs parallel.
  4. At no point of intersection are the tangents to the graphs perpendicular.

How to recognise this question

Question type: inverse functions

  • The command and notation point to inverse functions.
  • The requested response is a multiple choice.

How to handle it: replace the function value by y, swap x and y, then solve for y.

Watch out: Do not select a formula until its domain, interval, sign and units match the question.

Step-by-step answer

  1. A and B are false: intersections may lie on (e.g. fixed points) and may also occur in pairs off for some functions.
  2. C is false: tangents can be parallel (e.g. both of slope 1 at a fixed point on ).
  3. At an intersection point, if the tangents were perpendicular their slopes would satisfy .
  4. Using and the geometry of inverse graphs (reflection in ), the product of the tangent slopes at a common point cannot be under the global differentiability hypothesis of the question.
  5. Equivalently: slopes that are reciprocals related by the inverse-function derivative identity are inconsistent with a product of at every intersection configuration allowed here.
  6. Answer: D. (NESA multiple-choice key: D.)

Common mistake: choosing a familiar formula before checking that its conditions, signs, interval and units match this question.

The paper records 1 total mark for this group. Where the official marking guide provides useful partial-credit criteria, those criteria are stated in the steps above.

Try your hand at it again by answering these questions

Question 1

If and intersect at on and , the slope of at that point is

See the step-by-step answer guide →

Question 2

At a fixed-point intersection with slope for , the product of the two tangent slopes is

See the step-by-step answer guide →

Question 3

Can the tangents to and be parallel at an intersection on ?

  1. Yes, when , i.e.
  2. Never
  3. Only when
  4. Only when the functions are constant
See the step-by-step answer guide →