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 (iv) Graph of an inverse or open all working together.

(v)

Fixed points

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.

Explore this idea

Hint

  1. For a decreasing function the curves y=f(x)y=f(x) and y=f−1(x)y=f^{-1}(x) meet on y=xy=x.

  2. So solve f(x)=xf(x)=x.

Answer · (v)

f(x)=x⇒25x3−10x2+2x−2=0f(x)=x\Rightarrow25x^3-10x^2+2x-2=0.

For the restricted function; general decreasing functions can have additional inverse-intersection two-cycles.

Explore the idea

Open interactive exploration

f meets y = x at one point

The curves f and f⁻¹ cross where f meets the line y = x.

Switch on a feature to add it to the sketch and see how it is found. A full-mark sketch shows all of them.

  • Restricted f
  • Y = x
  • Intersection
Ideas to try
Why is the cubic's only real root the right one?