2017 H2 Mathematics Paper 2 Question 3

Functions

Answers

(ai)
(a2,0),(0,b){\left(\frac{a}{2},0\right), (0,b)}
(aii)
(a+1,0){\left(a+1,0\right)}
(aiii)
(a+12,0){\left(\frac{a+1}{2},0\right)}
(aiv)
(0,a),(b,0){\left(0, a\right), (b,0)}
(bi)
a=1{a=1} since g(1){g(1)} is undefined
(bii)
g2(x)=x{g^2(x)=x}
g−1(x)=1−11−x{g^{-1}(x) = 1-\frac{1}{1-x}}
(biii)
b=0   or   b=2{b=0 \; \textrm{ or } \; b=2}

Full solutions

(ai)
We apply a scaling of scale factor 12{\frac{1}{2}} parallel to the x-{x\textrm{-}}axis on y=f(x){y=f(x)}
(a2,0)  ■(0,b)  ■\begin{gather*} \left(\frac{a}{2},0\right) \; \blacksquare \\ (0,b) \; \blacksquare \end{gather*}
(aii)
We apply a translation of 1{1} unit along the positive x-{x\textrm{-}}axis direction on y=f(x){y=f(x)}
(a+1,0)  ■\left(a+1,0\right) \; \blacksquare
(aiii)
We apply a scaling of scale factor 12{\frac{1}{2}} parallel to the x-{x\textrm{-}}axis on y=f(x−1){y=f(x-1)}
(a+12,0)  ■\left(\frac{a+1}{2},0\right) \; \blacksquare
(aiv)
(0,a)  ■(b,0)  ■\begin{gather*} \left(0,a\right) \; \blacksquare \\ (b,0) \; \blacksquare \end{gather*}
(bi)
a=1  ■a=1 \; \blacksquare
since g(1){g(1)} is undefined ■{\blacksquare}
(bii)
g2(x)=g(1−11−x)=1−11−(1−11−x)=1−111−x=1−(1−x)=x  ■\begin{align*} g^2(x) &= g\left( 1 - \frac{1}{1-x} \right) \\ &= 1 - \frac{1}{1-\left( 1 - \frac{1}{1-x} \right)} \\ &= 1 - \frac{1}{\frac{1}{1-x}} \\ &= 1 - (1-x) \\ &= x \; \blacksquare \end{align*}
g2(x)=xgg(x)=xg(x)=g−1(x)g−1(x)=1−11−x  ■\begin{align*} g^2(x) &= x \\ gg(x) &= x \\ g(x) &= g^{-1}(x) \\ g^{-1}(x) &= 1 - \frac{1}{1-x} \; \blacksquare \end{align*}
(biii)
g2(b)=g−1(b)b=1−11−bb−b2=1−b−12b−b2=0b(2−b)=0\begin{gather*} g^2(b) = g^{-1}(b) \\ b = 1 - \frac{1}{1-b} \\ b - b^2 = 1-b-1 \\ 2b - b^2 = 0 \\ b(2-b) = 0 \end{gather*}
b=0  ■or b=2  ■\begin{align*} && b &= 0 \; \blacksquare \\ \textrm{or } && b &=2 \; \blacksquare \end{align*}