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Question -

If┬аf(x) = sin┬аx┬аand┬аg(x) = 2x be two real functions,then describe┬аgof┬аand┬аfog.┬аAre these equal functions?



Answer -

Given f(x) =sin┬аx┬аand┬аg(x) = 2x

We┬аknow┬аthat

f:┬аRтЖТ┬а[тИТ1,┬а1]┬аand┬аg:┬аRтЖТ┬аR

Clearly,┬аthe┬аrange┬аof┬аf┬аis┬аa┬аsubset┬аof┬аthe┬аdomain┬аof┬аg.

gof:┬аRтЖТ┬аR

(gof)┬а(x)┬а=┬аg┬а(f┬а(x))

=┬аg┬а(sin┬аx)

=┬а2┬аsin┬аx

Clearly,┬аthe┬аrange┬аof┬аg┬аis┬аa┬аsubset┬аof┬аthe┬аdomain┬аof┬аf.

fog:┬аR┬атЖТ┬аR

So,┬а(fog)┬а(x)┬а=┬аf┬а(g┬а(x))

=┬аf┬а(2x)

=┬аsin┬а(2x)

Clearly,┬аfog тЙаgof

Hence they are notequal functions.

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