Question
If $$f\left( x \right) = ax + b$$ and $$g\left( x \right) = cx + d,$$ then $$f\left\{ {g\left( x \right)} \right\} = g\left\{ {f\left( x \right)} \right\}$$ is equivalent to :
A.
$$f\left( a \right) = g\left( c \right)$$
B.
$$f\left( b \right) = g\left( b \right)$$
C.
$$f\left( d \right) = g\left( b \right)$$
D.
$$f\left( c \right) = g\left( a \right)$$
Answer :
$$f\left( d \right) = g\left( b \right)$$
Solution :
$$\eqalign{
& {\text{Given, }} \cr
& f\left( x \right) = ax + b,\,g\left( x \right) = cx + d{\text{ and }}f\left\{ {g\left( x \right)} \right\} = g\left\{ {f\left( x \right)} \right\} \cr
& \Rightarrow f\left( {cx + d} \right) = g\left( {ax + b} \right) \cr
& \Rightarrow a\left( {cx + d} \right) + b = c\left( {ax + b} \right) + d \cr
& \Rightarrow acx + ad + b = cax + bc + d \cr
& \Rightarrow ad + b = bc + d \cr
& \Rightarrow f\left( d \right) = g\left( b \right) \cr} $$