Question -
Answer -
(i)Given, f: R → R and g: R → R
So, gof: R → R and fog: R → R
Also given that f(x) =2x + 3 and g(x) = x2 + 5
Now, (gof) (x) = g (f (x))
= g (2x +3)
= (2x + 3)2 + 5
= 4x2+ 9 + 12x +5
=4x2+ 12x + 14
Now, (fog) (x) = f (g(x))
= f (x2 + 5)
= 2 (x2 + 5) +3
= 2 x2+ 10 + 3
= 2x2 + 13
(ii)Given, f: R → R and g: R → R
so, gof: R → R and fog: R → R
f(x) =2x + x2 and g(x) = x3
(gof) (x)= g (f (x))
= g (2x+x2)
= (2x+x2)3
Now, (fog) (x)= f (g (x))
= f (x3)
= 2 (x3)+ (x3)2
= 2x3 +x6
(iii)Given, f: R → R and g: R → R
So, gof: R → R and fog: R → R
f(x) = x2 +8 and g(x) = 3x3 + 1
(gof) (x)= g (f (x))
= g (x2 + 8)
= 3 (x2+8)3 + 1
Now, (fog) (x)= f (g (x))
= f (3x3 + 1)
= (3x3+1)2 + 8
= 9x6 + 6x3 + 1+ 8
= 9x6 +6x3 + 9
(iv)Given, f: R → R and g: R → R
So, gof: R → R and fog: R → R
f(x)= x and g(x) = |x|
(gof) (x)= g (f (x))
= g (x)
= |x|
Now (fog) (x)= f (g (x))
= f (|x|)
= |x|
(v)Given, f: R → R and g: R → R
So, gof: R → R and fog: R → R
f(x) = x2 +2x − 3 and g(x) = 3x − 4
(gof) (x)= g (f(x))
= g (x2 +2x − 3)
= 3 (x2 +2x − 3) − 4
= 3x2 + 6x − 9 − 4
= 3x2 +6x − 13
Now, (fog) (x)= f (g (x))
= f (3x − 4)
= (3x − 4)2 +2 (3x − 4) −3
= 9x2 +16 − 24x + 6x – 8 − 3
= 9x2 −18x + 5
(vi)Given, f: R → R and g: R → R
So, gof: R → R and fog: R → R
f(x) = 8x3 and g(x)= x1/3
(gof) (x)= g (f (x))
= g (8x3)
= (8x3)1/3
= [(2x)3]1/3
= 2x
Now, (fog) (x)= f (g (x))
= f (x1/3)
= 8 (x1/3)3
= 8x