Question -
Answer -
(i)Given,┬аf:┬аR┬атЖТ┬аR┬аand┬аg:┬аR┬атЖТ┬аR
So,┬аgof:┬аR┬атЖТ┬аR┬аand┬аfog:┬аR┬атЖТ┬аR
Also given that f(x) =2x┬а+ 3 and┬аg(x) =┬аx2┬а+ 5
Now, (gof) (x) = g (f (x))
= g┬а(2x┬а+3)
=┬а(2x┬а+ 3)2┬а+ 5
=┬а4x2+ 9 + 12x┬а+5
=4x2+┬а12x┬а+ 14
Now, (fog) (x) = f (g(x))
= f (x2┬а+ 5)
=┬а2┬а(x2┬а+ 5) +3
=┬а2┬аx2+ 10 + 3
=┬а2x2┬а+ 13
(ii)Given,┬аf:┬аR┬атЖТ┬аR┬аand┬аg:┬аR┬атЖТ┬аR
so,┬аgof:┬аR┬атЖТ┬аR┬аand┬аfog:┬аR┬атЖТ┬аR
f(x) =2x┬а+┬аx2┬аand┬аg(x) =┬аx3
(gof)┬а(x)=┬аg┬а(f┬а(x))
=┬аg┬а(2x+x2)
=┬а(2x+x2)3
Now, (fog)┬а(x)=┬аf┬а(g┬а(x))
=┬аf┬а(x3)
=┬а2┬а(x3)+ (x3)2
= 2x3┬а+x6
(iii)Given,┬аf:┬аR┬атЖТ┬аR┬аand┬аg:┬аR┬атЖТ┬аR
So,┬аgof:┬аR┬атЖТ┬аR and┬аfog:┬аR┬атЖТ┬аR
f(x) =┬аx2┬а+8┬а and┬аg(x) = 3x3┬а+ 1
(gof)┬а(x)=┬аg┬а(f┬а(x))
=┬аg┬а(x2┬а+┬а8)
=┬а3┬а(x2+8)3┬а+┬а1
Now, (fog)┬а(x)=┬аf┬а(g┬а(x))
=┬аf┬а(3x3┬а+┬а1)
=┬а(3x3+1)2┬а+┬а8
=┬а9x6┬а+┬а6x3┬а+┬а1+┬а8
= 9x6┬а+6x3┬а+ 9
(iv)Given,┬аf:┬аR┬атЖТ┬аR┬аand┬аg:┬аR┬атЖТ┬аR
So,┬аgof:┬аR┬атЖТ┬аR┬аand┬аfog:┬аR┬атЖТ┬аR
f(x)=┬аx┬аand┬аg(x) = |x|
(gof)┬а(x)=┬аg┬а(f┬а(x))
=┬аg┬а(x)
=┬а|x|
Now (fog)┬а(x)=┬аf┬а(g┬а(x))
=┬аf┬а(|x|)
=┬а|x|
(v)Given,┬аf:┬аR┬атЖТ┬аR┬аand┬аg:┬аR┬атЖТ┬аR
So,┬аgof:┬аR┬атЖТ┬аR┬аand┬аfog:┬аR┬атЖТ┬аR
f(x) =┬аx2┬а+2x┬атИТ 3 and┬аg(x) = 3x┬атИТ 4
(gof)┬а(x)=┬аg┬а(f(x))
=┬аg┬а(x2┬а+2x тИТ 3)
=┬а3┬а(x2┬а+2x тИТ 3)┬атИТ 4
=┬а3x2┬а+┬а6x┬атИТ┬а9┬атИТ┬а4
=┬а3x2┬а+6x тИТ 13
Now, (fog)┬а(x)=┬аf┬а(g┬а(x))
=┬аf┬а(3x тИТ 4)
=┬а(3x┬атИТ┬а4)2┬а+2┬а(3x┬атИТ┬а4)┬атИТ3
=┬а9x2┬а+16 тИТ 24x + 6x тАУ 8 тИТ 3
=┬а9x2┬атИТ18x┬а+┬а5
(vi)Given,┬аf:┬аR┬атЖТ┬аR┬аand┬аg:┬аR┬атЖТ┬аR
So,┬аgof:┬аR┬атЖТ┬аR┬аand┬аfog:┬аR┬атЖТ┬аR
f(x) = 8x3┬аand┬аg(x)=┬аx1/3
(gof)┬а(x)=┬аg┬а(f┬а(x))
=┬аg┬а(8x3)
= (8x3)1/3
= [(2x)3]1/3
=┬а2x
Now, (fog)┬а(x)=┬аf┬а(g┬а(x))
= f (x1/3)
= 8 (x1/3)3
=┬а8x