Let f(x)=ax+b, x∈R, and g(x)=x+d, x∈R, then fog=gof if and only if

f(g(x))=g(f(x)) for all x∈R⇔f(cx+d)=g(ax+b)⇔ a(cx+d)+b=c(ax+b)+d⇔ad+b=cb+d⇔ f(d)=g(b)

Let f(x)=ax+b, x∈R, and g(x)=x+d, x∈R, then fog=gof if and only if

f(g(x))=g(f(x)) for all x∈R⇔f(cx+d)=g(ax+b)⇔ a(cx+d)+b=c(ax+b)+d⇔ad+b=cb+d⇔ f(d)=g(b)

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Let $f (x) = a x + b, x \in R, a n d g (x) = x + d, x \in R,$ then $f o g = g o f$ if and only if

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$f (a) = g (c)$

$f (d) = g (b)$

$f (b) = g (d)$

$f (c) = g (a)$

answer is B.

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$f (g (x)) = g (f (x))$ for all $x \in R \Leftrightarrow f (c x + d) = g (a x + b)$
$\Leftrightarrow a (c x + d) + b = c (a x + b) + d \Leftrightarrow a d + b = c b + d$
$\Leftrightarrow f (d) = g (b)$

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Let f(x)=ax+b, x∈R, and g(x)=x+d, x∈R, then fog=gof if and only if

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Let $f (x) = a x + b, x \in R, a n d g (x) = x + d, x \in R,$ then $f o g = g o f$ if and only if