Invert, then repeat periodically

(a) f(x)=ex+3−af(x)=e^{x+3}-a for real x≤4x\le4; g(x)=2x+5g(x)=2x+5 for all real x. The parameter a is constant.

(i) Find the inverse of f and its domain in terms of a. marks

(ii) Given that the composite exists and g−1f(−3)=15g^{-1}f(-3)=15, find a without finding the composite rule. marks

(b) h(x)=2xh(x)=2^x for 0<x≤20<x\le2, and h(x)=6−x2/2h(x)=6-x^2/2 for 2<x≤32<x\le3, with h(x)=h(x−3)h(x)=h(x-3) for all real x.

(i) Find h(9). mark

(ii) Sketch h on [−1,4][-1,4]. marks

[2023/ACJC/Promo/5]

Total marks: 10

Solution

(a)i

Inverse functions

f(x)=ex+3−af(x)=e^{x+3}-a for real x≤4x\le4, where aa is a constant. Find f−1f^{-1} and its domain in terms of aa.

Hint

  1. Let y=f(x)y=f(x) and solve for xx.

  2. Swap xx and yy.

  3. The domain of f−1f^{-1} is the range of ff, found from the endpoints of the domain of ff.

Answer · (a)i

f−1(x)=ln⁡(x+a)−3f^{-1}(x)=\ln(x+a)-3, −a<x≤e7−a-a<x\le e^7-a.

(a)ii

Using an inverse equation

g(x)=2x+5g(x)=2x+5 for all real xx. Given that the composite g−1fg^{-1}f exists and g−1f(−3)=15g^{-1}f(-3)=15, find aa without finding the composite rule.

Hint

  1. Apply gg to both sides of g−1f(−3)=15g^{-1}f(-3)=15.

  2. g(g−1(u))=ug\bigl(g^{-1}(u)\bigr)=u, so f(−3)=g(15)f(-3)=g(15).

  3. Substitute the rules.

Answer · (a)ii

a=−34a=-34.

(b)i

Periodic functions

h(x)=2xh(x)=2^x for 0<x≤20<x\le2 and h(x)=6−x22h(x)=6-\frac{x^2}2 for 2<x≤32<x\le3, with h(x)=h(x−3)h(x)=h(x-3) for all real xx. Find h(9)h(9).

Hint

  1. hh has period 3: subtract multiples of 3 until the input is in (0,3](0,3].

Answer · (b)i

h(9)=3/2h(9)=3/2.

(b)ii

Sketching a periodic piecewise function

Sketch y=h(x)y=h(x) for −1≤x≤4-1\le x\le4, labelling the end-points.

Explore this idea

Hint

  1. Sketch one period (0,3](0,3]: the exponential on (0,2](0,2] then the quadratic on (2,3](2,3].

  2. Copy it 3 units left and right.

  3. Mark open and closed ends carefully.

Periodic h on −1≤x≤4
Periodic h on −1≤x≤4Scroll across to see the full diagram.

Answer · (b)ii

Period 3; closed values (−1,4),(0,3/2),(2,4),(3,3/2),(4,2)(-1,4),(0,3/2),(2,4),(3,3/2),(4,2); open (0,1),(3,1)(0,1),(3,1).

Explore the idea

A piecewise graph that repeats every 3

Switch on the closed and open end-points.

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

  • Y = h(x)
  • Period boundaries
Ideas to try
Why does the graph have a jump at x=0x=0 and at x=3x=3?