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Is It True What These Students Claim...?

In 1972, the journal Mathematics Teaching published this Letter to the Editor, sent by two secondary students.

Dear sir,
On investigating graphs of functions, it was found that if the scale of graphs xn was changed at the point x=1 then the curve continued as if the scale had not been altered! (Please see the enclosed example).

This only works when the x and y scales are exactly the same.

We were advised to put this to you for further investigation, by our Maths teacher. We hope you will find it as interesting as we did.

Yours faithfully,
Queen Elizabeth's School
Crediton, Devon

Task 1: Are these two students correct in their claim?

Task 2: What happens to their claim if you shift to any function...perhaps keeping the requirement that its graph pass through the coordinate point (1,1)?


Source: Mathematics Teaching, Summer 1972, pp. 48-49

Hint: Draw many examples...try different scales for x>1, both greater and less than the scale for the interval [0,1].

Try n=1, 2, 3, 4, ... or even n = 1/2, etc.


Solution Commentary: What did you learn or discover?

Is the claim provable...other than increased confidence in its veracity through examples?