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1.7.1 Function, SPM Practice (Long Question)


Question 1:
The function f and g is defined by
f:x2x3g:x2x;x0
Find the expression for each of the following functions
(a) ff,
(b) gf,
(c) f-1 ,
Calculate the value of x such that ff(x) = gf(x).

Solution:
(a)
ff(x)=f[f(x)]         =f(2x3)         =2(2x3)3         =4x9ff:x4x9

(b)
gf(x)=g[f(x)]          =g(2x3)          =22x3gf:x22x3

(c)
Let f1(x)=y,thus  f(y)=x      2y3=x              y=x+32  f1(x)=x+32f1:xx+32When ff(x)=gf(x),4x9=22x3(4x9)(2x3)=28x230x+27=28x230x+25=0(4x5)(2x5)=04x5=0       or       2x5=0x=54             or              x=52

Question 2:
The function f and g is defined by
f(x)=3x2g(x)=3x,x0Find(a) f1(2),(b) gf(3),(c) function h if hf(x)=3x+2,(d) function k if fk(x)=4x7.

Solution:
(a)
Let f1(2)=x,thus  f(x)=2      3x2=2           3x=4             x=43f1(2)=43

(b)
gf(3)=g[3(3)2]           =g(11)           =311

(c)
h[f(x)]=3x+2h(3x2)=3x+2Let y=3x2thus     x=y+23     h(y)=3(y+23)+2            =y+2+2            =y+4 h(x)=x+4

(d)
f[k(x)]=4x73k(x)2=4x73k(x)=4x5k(x)=4x53

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