First slide

Functions (XII)

Question

$Let f (x) = a x + b and g (x) = c x + d, a \neq 0, c \neq 0 . Assume a = 1, b = 2 . If (f \circ g) (x) = (g o f) (x) for all x, what can you$
$say about c and d ?$

Moderate

A

c and d both arbitary
B

c = 1, d arbitrary
C

c arbitrary, d
D

c=1,d=1

Solution

$(f o g) (x) = f (g (x)) = a (c x + d) + b$
$and (g o f) (x) = f (f (x)) = c (a x + b) + d$
$Given that (f o g) (x) = (g o f) (x) and at a = 1, b = 2$
$\Rightarrow c x + d + 2 = c x + 2 c + d \Rightarrow c = 1 and d is arbitary$

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