Let \(f(x) = \left| x \right| - 1\).

(a)     The graph of \(y = g(x)\) is drawn below.

 

          (i)     Find the value of \((f \circ g)(1)\).

          (ii)     Find the value of \((f \circ g \circ g)(1)\).

          (iii)     Sketch the graph of \(y = (f \circ g)(x)\).

(b)     (i)     Sketch the graph of \(y = f(x)\).

          (ii)     State the zeros of f.

(c)     (i)     Sketch the graph of \(y = (f \circ f)(x)\).

          (ii)     State the zeros of \(f \circ f\).

(d)     Given that we can denote \(\underbrace {f \circ f \circ f \circ  \ldots  \circ f}_{n{\text{ times}}}\) as \({f^n}\),

          (i)     find the zeros of \({f^3}\);

          (ii)     find the zeros of \({f^4}\);

          (iii)     deduce the zeros of \({f^8}\).

(e)     The zeros of \({f^{2n}}\) are \({a_1},{\text{ }}{a_2},{\text{ }}{a_3},{\text{ }} \ldots {\text{, }}{a_N}\).

          (i)     State the relation between n and N;

          (ii)     Find, and simplify, an expression for \(\sum\limits_{r = 1}^N {\left| {{a_r}} \right|} \) in terms of n.

(a)     (i)     \(f(0) =  - 1\)     (M1)A1

          (ii)     \((f \circ g)(0) = f(4) = 3\)     A1

          (iii)
               (M1)A1

 

Note:     Award M1 for evidence that the lower part of the graph has been reflected and A1 correct shape with y-intercept below 4.

 

[5 marks]

 

(b)     (i)
               (M1)A1

 

Note:     Award M1 for any translation of \(y = \left| x \right|\).

 

          (ii)     \( \pm 1\)     A1

 

Note:     Do not award the A1 if coordinates given, but do not penalise in the rest of the question

 

[3 marks]

 

(c)     (i)
               (M1)A1

 

Note:     Award M1 for evidence that lower part of (b) has been reflected in the x-axis and translated.

 

          (ii)     \(0,{\text{ }} \pm 2\)     A1

[3 marks]

 

(d)     (i)     \( \pm 1,{\text{ }} \pm 3\)     A1

          (ii)     \(0,{\text{ }} \pm 2,{\text{ }} \pm 4\)     A1

          (iii)     \(0,{\text{ }} \pm 2,{\text{ }} \pm 4,{\text{ }} \pm 6,{\text{ }} \pm 8\)     A1

[3 marks]

 

(e)     (i)     \({\text{(1, 3), (2, 5), }} \ldots \)     (M1)

          \(N = 2n + 1\)     A1

          (ii)     Using the formula of the sum of an arithmetic series     (M1)

          EITHER

          \(4(1 + 2 + 3 +  \ldots  + n) = \frac{4}{2}n(n + 1)\)

          \( = 2n(n + 1)\)     A1

          OR

          \(2(2 + 4 + 6 +  \ldots  + 2n) = \frac{2}{2}n(2n + 2)\)

          \( = 2n(n + 1)\)     A1

[4 marks]

 

Total [18 marks]

[N/A]