Compose through a restricted range, then use symmetry for area

(a) Functions f:x↦(x−2)(x−4) f:x\mapsto(x-2)(x-4) for x∈Rx\in\mathbb R, x<4 x<4; g:x↦x/(x+3) g:x\mapsto x/(x+3) for x∈Rx\in\mathbb R, x≠−3 x\ne-3.

Determine if ff is one-one, justifying your answer. mark

Only one of fgfg and gfgf exists. Give a definition (including domain) of the composite that exists, and explain why the other does not exist. Find the range of the composite that exists. marks

(b) h:x↦a(1+e−x) h:x\mapsto a(1+e^{-x}), x∈R0+ x\in\mathbb R_0^+, where aa is positive. Find h−1(x) h^{-1}(x) and its domain. marks

Given that hh and h−1 h^{-1} intersect where x=bx=b, and ∫0bh(x) dx=I\int_0^b h(x)\, dx=I , find the area bounded by h,h−1 h, h^{-1} and the axes in terms of I,bI, b. marks

[2010/TJC/Prelim/II/3]

Total marks: 12

Solution

If this part uses an earlier result, revisit (a) composite Existence and range of composites or open all working together.

(b) inverse

Inverse functions

h(x)=a(1+e−x)h(x)=a(1+e^{-x}) for x>0x>0, where a>0a>0. Find h−1(x)h^{-1}(x) and its domain.

Explore this idea

Hint

  1. Write y=a(1+e−x)y=a(1+e^{-x}) and make xx the subject.

  2. Dh−1=RhD_{h^{-1}}=R_h.

Answer · (b) inverse

h−1(x)=−ln⁡(x/a−1)h^{-1}(x)=-\ln(x/a-1).

If ℝ₀⁺ means x≥0: inverse domain (a,2a]. If intended x>0 as solution uses: (a,2a). Endpoint ambiguity remains explicit.

Explore the idea

Open interactive exploration

h and its inverse for a = 2

Each curve has an asymptote that is the mirror of the other's: y = a for h, and x = a for h⁻¹.

Point on f: input xx = 1.00
Reset the controls

Reflection swaps the coordinates. The segment PQ is perpendicular to y = x, and its blue midpoint M lies on that line.

P = (x, f(x))
(1, 2.73576)
Q = (f(x), x)
(2.73576, 1)
on the inverse
Domain of f
x≥0x\ge0
Range of f = domain of f⁻¹
a<y≤2aa<y\le2a
Midpoint M on y = x
(1.86788, 1.86788)
  • H(x) = 2(1 + e⁻ˣ)
  • H⁻¹
  • Y = x, where h and h⁻¹ meet
  • PQ is perpendicular to y = x; its blue midpoint M lies on that line
Key idea

The two curves meet exactly on y=xy=x, at the point (b,b)(b,b) used in the area question.

Ideas to try
Where do hh and h−1h^{-1} meet? What is that xx called in the question?