Question 1:
The function
f and
g is defined by
f:x↦2x−3g:x↦2x;x≠0
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(2x−3) =2(2x−3)−3 =4x−9ff:x↦4x−9
(b)
gf(x)=g[f(x)] =g(2x−3) =22x−3gf:x↦22x−3
(c)
Let f−1(x)=y,thus f(y)=x 2y−3=x y=x+32∴ f−1(x)=x+32f−1:x↦x+32When ff(x)=gf(x),4x−9=22x−3(4x−9)(2x−3)=28x2−30x+27=28x2−30x+25=0(4x−5)(2x−5)=04x−5=0 or 2x−5=0x=54 or x=52
Question 2:
The function
f and
g is defined by
f(x)=3x−2g(x)=3x,x≠0Find(a) f−1(2),(b) gf(−3),(c) function h if hf(x)=3x+2,(d) function k if fk(x)=4x−7.
Solution:
(a)
Let f−1(2)=x,thus f(x)=2 3x−2=2 3x=4 x=43f−1(2)=43
(b)
gf(−3)=g[3(−3)−2] =g(−11) =−311
(c)
h[f(x)]=3x+2h(3x−2)=3x+2Let y=3x−2thus x=y+23 h(y)=3(y+23)+2 =y+2+2 =y+4∴ h(x)=x+4
(d)
f[k(x)]=4x−73k(x)−2=4x−73k(x)=4x−5k(x)=4x−53