Find gof and fog when f: R → R and g: R → R is defined by f(x)

1.	Find gof and fog when f: R → R and g: R → R is defined by f(x) = x2 + 2x – 3 and g(x) = 3x – 4
Answer» Since, f:R → R and g:R → R fog:R → R and gof:R → R f(x) = x² + 2x – 3 and g(x) = 3x – 4 Now, gof(x)=g(f(x))= g(x² + 2x – 3) gof(x) = 3(x² + 2x–3) – 4 ⇒ gof(x)= 3x² + 6x – 9 – 4 ⇒ gof(x) = 3x² + 6x – 13 and, fog= f(g(x)) = f(3x – 4) fog(x) = (3x – 4)² + 2(3x – 4) – 3 = 9x² + 16 – 24x + 6x – 8 – 3 ∴ fog(x) = 9x² – 18x + 5 Thus, gof(x) = 3x² + 6x – 13 and fog(x) = 9x² – 18x + 5

Find gof and fog when f: R → R and g: R → R is defined by f(x) = x2 + 2x – 3 and g(x) = 3x – 4

Answer»

Since, f:R → R and g:R → R

fog:R → R and gof:R → R

f(x) = x² + 2x – 3 and g(x) = 3x – 4

Now, gof(x)=g(f(x))= g(x² + 2x – 3)

gof(x) = 3(x² + 2x–3) – 4

⇒ gof(x)= 3x² + 6x – 9 – 4

⇒ gof(x) = 3x² + 6x – 13

and, fog= f(g(x)) = f(3x – 4)

fog(x) = (3x – 4)² + 2(3x – 4) – 3

= 9x² + 16 – 24x + 6x – 8 – 3

∴ fog(x) = 9x² – 18x + 5

Thus, gof(x) = 3x² + 6x – 13 and fog(x) = 9x² – 18x + 5