A non-invertible function can have an invertible composite

f:x↦(x−7)(x+1) f:x\mapsto(x-7)(x+1), x>0 x>0.

(a) Use an algebraic method to find exact solution of f(x)<(x2−7x)/(2−x) f(x)<(x^2-7x)/(2-x). marks

(b)(i) Determine if f−1 f^{-1} exists. marks

Given gg is one-one with domain (0,∞)(0, \infty) and range (4,∞)(4, \infty):

(ii) Find range of fgfg. marks

(iii) A function hh is one-one if for any α,β∈Dh\alpha, \beta\in D_h , h(α)=h(β) h(\alpha)=h(\beta) implies α=β\alpha=\beta . Show that fgfg is one-one. marks

(iv) If g(x)=4exg(x)=4e^x for x>0 x>0, find exact (fg)−1(20)(fg)^{-1}(20). marks

[2011/RI/Promo/13]

Total marks: 15

Solution

If this part uses an earlier result, revisit (b)i One-one functions or open all working together.

(b)ii

Range of a composite

gg is one-one with domain (0,∞)(0,\infty) and range (4,∞)(4,\infty). Find the range of fgfg.

Explore this idea

Hint

  1. Rg=(4,∞)R_g=(4,\infty) is the restricted domain for ff.

  2. Read the range of ff on (4,∞)(4,\infty) from the graph.

Answer · (b)ii

Rfg=(−15,∞)R_{fg}=(-15,\infty).

Explore the idea

Open interactive exploration

Only the right-hand arm of f is used

g feeds f only numbers above 4, so only the increasing part of the parabola counts.

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

  • F on (4, ∞)
Ideas to try
What would the range be if RgR_g were (2,∞)(2,\infty)?