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Achievement objective M8-2

In a range of meaningful contexts, students will be engaged in thinking mathematically and statistically. They will solve problems and model situations that require them to:

  • display and interpret the graphs of functions with the graphs of their inverse and/or reciprocal functions.


  • Demonstrates understanding of inverse and/or reciprocal functions, including those of:
    • polynomial
    • exponential y = eax or y = bx
    • logarithmic
    • rectangular hyperbola y equals a divided by bracket x plus or minus b bracket.
    • trigonometric equations of the form y = Asin(Bx ) + C for sin, cos and tan functions with x in radians or degrees
    • f(x) = √ x
  • Makes graphical and algebraic connections between a function and its inverse, including the consideration of restrictions on the domain of a function so that its inverse is also a function.
  • Makes links with manipulate trigonometric expressions M8-6, solving equations M8-7 and identification of discontinuities M8-10.
  • See key mathematical ideas on NZmaths.


M8-2 links from M7-2.

Possible context elaborations

  • Given the graph of ex, sketch its inverse function.
  • Given a function algebraically, find the rule for its inverse.
  • Explore the reciprocals and inverses of trigonometric functions.
  • Investigate families of functions and their inverses making connections between transformations of a graph and changes to the equation of the graph.
  • Use equations of functions to model situations such as temperature data, tides, ferris wheels, ski slopes, population growth.
  • Activity: Big Bang Theory - Fractals and complex numbers
  • Activity: Linking integration and differentiation

Assessment for qualifications

NCEA achievement standards at levels 1, 2 and 3 have been aligned to the New Zealand Curriculum. Please ensure that you are using the correct version of the standards by going to the NZQA website.

The NZQA subject-specific resources pages are very helpful. From there, you can find all the achievement standards and links to assessment resources, both internal and external.

Learn more:

The following achievement standard(s) could assess learning outcomes from this AO:

  • There are currently no achievement standards that could assess learning outcomes from this AO.

Refer to the mathematics and statistics matrix.

Last updated September 26, 2013