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NDA 2017
Exam
NDA
Year
2017
Subject
Mathematics
Chapter
Functions

NDA 2017 · Mathematics · Functions

Let f(x)=px+q and g(x)=mx+n . Then f( g(x))=g(f(x)) is equivalent to

Let f(x)=px+qf(\mathrm{x})=\mathrm{px}+\mathrm{q} and g(x)=mx+n\mathrm{g}(\mathrm{x})=\mathrm{mx}+\mathrm{n}. Then f(g(x))=g(f(x))f(\mathrm{ g}(\mathrm{x}))=\mathrm{g}(f(\mathrm{x})) is equivalent to

Options

  1. Af(p)=g(m)f(\mathrm{p})=\mathrm{g}(\mathrm{m})
  2. Bf(q)=g(n)f(\mathrm{q})=\mathrm{g}(\mathrm{n})
  3. Cf(n)=g(q)f(\mathrm{n})=\mathrm{g}(\mathrm{q})
  4. Df(m)=g(p)f(\mathrm{ m})=\mathrm{g}(\mathrm{p})

Correct answer

C. f(n)=g(q)f(\mathrm{n})=\mathrm{g}(\mathrm{q})

Step-by-step solution

f(x)=px+q,g(x)=mx+nf(g(x))=g(f(x))⇒f(mx+n)=g(px+q)⇒p(mx+n)+q=m(px+q)+n⇒pmx+pn+q=pmx+mq+n⇒pn+q=mq+n⇒f(n)=g(q)\begin{aligned} & \mathrm{f}(\mathrm{x})=\mathrm{px}+\mathrm{q}, \mathrm{g}(\mathrm{x})=\mathrm{mx}+\mathrm{n} \\ & \mathrm{f}(\mathrm{g}(\mathrm{x}))=\mathrm{g}(\mathrm{f}(\mathrm{x})) \\ & \Rightarrow \mathrm{f}(\mathrm{mx}+\mathrm{n})=\mathrm{g}(\mathrm{px}+\mathrm{q}) \\ & \Rightarrow \mathrm{p}(\mathrm{mx}+\mathrm{n})+\mathrm{q}=\mathrm{m}(\mathrm{px}+\mathrm{q})+\mathrm{n} \\ & \Rightarrow \mathrm{pmx}+\mathrm{pn}+\mathrm{q}=\mathrm{pmx}+\mathrm{mq}+\mathrm{n} \\ & \Rightarrow \mathrm{pn}+\mathrm{q}=\mathrm{mq}+\mathrm{n} \\ & \Rightarrow \mathrm{f}(\mathrm{n})=\mathrm{g}(\mathrm{q})\end{aligned}
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