If f (x) = ax + b and g (x) = cx + d, then f g (x) = g f (x) is equivalent to

Question

If f (x) = ax + b and g (x) = cx + d, then f g (x) = g f (x) is equivalent to (A) f (a) = g (c) (B) f (b) = g (b) (C) f (d) = g (b) (D) f (c) = g (a)

Tardigrade Admin · Accepted Answer

We have f(x)=a x+b, g(x)=c x+d Therefore, f g ( x ) = g f ( x ) Leftrightarrow f ( cx + d )= g ( ax + b ) Leftrightarrow a(c x+d)+b=c(a x+b)+d Leftrightarrow ad + b = cb + d Leftrightarrow f ( d )= g ( b )

Q. If f (x) = ax + b and g (x) = cx + d, then f {g (x)} = g {f (x)} is equivalent to

f (a) = g (c)

f (b) = g (b)

f (d) = g (b)

f (c) = g (a)

We have $f (x) = a x + b, g (x) = c x + d$
Therefore, $f {g (x)} = g {f (x)} \Leftrightarrow f (c x + d) = g (a x + b)$
$< b r / >\Leftrightarrow a (c x + d) + b = c (a x + b) + d < b r / >$
$< b r / >\Leftrightarrow a d + b = c b + d \Leftrightarrow f (d) = g (b) < b r / >$

Q. If f (x) = ax + b and g (x) = cx + d, then f {g (x)} = g {f (x)} is equivalent to

f (a) = g (c)

f (b) = g (b)

f (d) = g (b)

f (c) = g (a)

We have f(x)=ax+b,g(x)=cx+d Therefore, f{g(x)}=g{f(x)}⇔f(cx+d)=g(ax+b) <br/>⇔a(cx+d)+b=c(ax+b)+d<br/> <br/>⇔ad+b=cb+d⇔f(d)=g(b)<br/>

We have $f (x) = a x + b, g (x) = c x + d$
Therefore, $f {g (x)} = g {f (x)} \Leftrightarrow f (c x + d) = g (a x + b)$
$< b r / >\Leftrightarrow a (c x + d) + b = c (a x + b) + d < b r / >$
$< b r / >\Leftrightarrow a d + b = c b + d \Leftrightarrow f (d) = g (b) < b r / >$