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NDA 2024

Exam: NDA
Year: 2024
Subject: Mathematics
Chapter: Functions

NDA 2024 · Mathematics · Functions

If f(x)=a x-b and g(x)=c x+d are such that f(g(x))=g(f(x)) , then...

If

f(x)=a x-b

and

g(x)=c x+d

are such that

f(g(x))=g(f(x))

, then which one of the following holds?

Options

A $f(d)=g(b)$
B $f(b)+g(d)=0$
C $f(a)+g(c)=2 a$
D $f(d)+g(b)=2 d$

Correct answer

D. $f(d)+g(b)=2 d$

Step-by-step solution

f(x)=a x-b \quad....(i)

and

\quad g(x)=c x+d \quad....(ii)

\begin{aligned} & f[g(x)]=g[f(x)] \\ & \Rightarrow \quad a[g(x)]-b=c[f(x)]+d \\ & \Rightarrow \quad a[c x+d]-b=c[a x-b]+d \\ & \text{ [using (i) and (ii)] } \\ & \Rightarrow \quad a c x+a d-b=a c x-b c+d \\ & \Rightarrow \quad a d-b=d-b c \\ & \Rightarrow \quad a d-b+b c+d=d+d \\ & \therefore \quad f(d)+g(b)=2 d \end{aligned}

$$

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