Restrict the domain to choose an inverse

For all real x, f(x)=2/[1+(5x−1)2]f(x)=2/[1+(5x-1)^2].

(i) Find its range. marks

(ii) Explain why it has no inverse. mark

(iii) Restrict its domain to x≥kx\ge k: find the least k that gives an inverse, and find that inverse. marks

(iv) Sketch the restricted f and its inverse together, clearly showing their relationship. marks

(v) Show algebraically that a solution of f(x)=f−1(x)f(x)=f^{-1}(x) satisfies 25x3−10x2+2x−2=025x^3-10x^2+2x-2=0. marks

[2011/IJC/Prelim/I/9]

Total marks: 10

Solution

If this part uses an earlier result, revisit (i) Range or open all working together.

(ii)

One-one functions

Explain why ff has no inverse.

Explore this idea

Hint

  1. Find two different inputs with the same output (horizontal-line test).

Answer · (ii)

No inverse: f(0)=f(2/5)=1f(0)=f(2/5)=1 with different inputs.

Explore the idea

Open interactive exploration

The horizontal line test

A function has an inverse only if every horizontal line meets its graph at most once.

Height of the horizontal linek = 1
Try an example
Test boundary heights

Buttons use full-precision boundary values; numbers shown here are rounded.

Number of distinct intersections
2

Two distinct inputs give this output, so the function is not one-one.

All listed intersections are within the original drawing window. A touching point counts once; hollow endpoints are excluded.

Coordinates are rounded to 6 significant figures. Inspect a row to highlight that intersection and bring it into view.
Pointxy
-2.22045e-171
0.41
  • F
  • Y = k
  • Intersections
  • Intersection selected in the coordinate table
Key idea

Two intersections at the same height mean two inputs share one output. The function is not one-one.

Ideas to try
For which kk does the line meet the graph exactly once?