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 (ii) One-one functions or open all working together.

(iii)

Restricting a domain

The domain of ff is restricted to x≥kx\ge k. Find the least kk for which f−1f^{-1} exists, and find f−1(x)f^{-1}(x).

Explore this idea

Hint

  1. The restricted piece must be monotonic: keep one side of the peak.

  2. Solve y=f(x)y=f(x) for xx and choose the branch that matches the domain.

  3. State the domain of the inverse.

Answer · (iii)

k=1/5k=1/5; f−1(x)=15(1+2/x−1)f^{-1}(x)=\frac15(1+\sqrt{2/x-1}), 0<x≤20<x\le2.

Explore the idea

Open interactive exploration

Cut the domain until an inverse exists

Move the left boundary k. Keep only x ≥ k and watch when the two branches become one.

Original question: find the least k for a domain x ≥ k. Reset restores the starting experiment; the peak preset gives the boundary case.

Left boundary k0

Closed endpoint: the boundary is included unless the original domain excludes it.

Try an example

Not one-to-one: at least one reached output comes from two retained inputs.

Proposed input domain

[0, ∞)

Proposed input domain-0.80.62

Retained inputs

[0, ∞)

Retained inputs-0.80.62

Reached outputs

(0, 2]

Reached outputs01.12.2

Original allowed inputs

(−∞, ∞)

These exclusions stay in force when a boundary changes. Arrows indicate continuation beyond the drawing window.

Both sides of the peak remain. A horizontal line can meet the retained graph twice.

The graph shows a finite window; arrows and interval notation describe the full set. Decimal endpoints are rounded.