First slide

Functions (XII)

Question

$If f (x) and g (x) are periodic functions with periods 7 and 11, respectively, then the period of F (x) = f (x) g (\frac{x}{5}) - g (x) f (\frac{x}{3}) is$

Moderate

A

177
B

222
C

433
D

1155

Solution

$T h e p e r i o d o f f (x) i s 7, s o t h e p e r i o d o f f (\frac{x}{3}) i s \frac{7}{\frac{1}{3}} = 21$ $\sin c e p e r i o d o f f (x) = p t h e n t h e p e r i o d o f f (a x + b) = \frac{p}{|a|}$
$\sin c e t h e p e r i o d o f g (x) = 11, t h e p e r i o d o f g (\frac{x}{5}) = \frac{11}{\frac{1}{5}} = 55 P_{1} = p e r i o d o f f (x) \cdot g (\frac{x}{5}) = 7 \cdot 55 = 385$
$a n d P_{2} = p e r i o d o f g (x) \cdot f (\frac{x}{3}) = 11 \cdot 21 = 231 t h e r e f o r e p e r i o d o f F (x) = L C M \{P_{1,} P_{2}\} = L C M \{385, 231\} = 1155$

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