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 + 8 and g(x) = 3x3 + 1
Answer» Since, f:R → R and g:R → R fog:R → R and gof:R → R f(x)=x² + 8 and g(x)=3x³ + 1 So, gof(x)= g(f(x)) gof(x)= g(x² + 8) gof(x)= 3(x² + 8)³ + 1 ⇒ gof(x)= 3(x⁶ + 512 + 24x⁴ + 192x²) + 1 ⇒ gof(x)= 3x⁶ + 72x⁴ + 576x² + 1537 Similarly, fog(x)=f(g(x)) ⇒ fog(x)= f(3x³ + 1) ⇒ fog(x)=(3x³ + 1)² + 8 ⇒ fog(x)=(9x⁶ + 1 + 6x³) + 8 ⇒ fog(x)=9x⁶ + 6x³ + 9 So, gof(x) = 3x⁶ + 72x⁴ + 576x² + 1537 and fog(x) = 9x⁶ + 6x³ + 9

Find gof and fog when f: R → R and g: R → R is defined by f(x) = x2 + 8 and g(x) = 3x3 + 1

Answer»

Since, f:R → R and g:R → R

fog:R → R and gof:R → R

f(x)=x² + 8 and g(x)=3x³ + 1

So, gof(x)= g(f(x))

gof(x)= g(x² + 8)

gof(x)= 3(x² + 8)³ + 1

⇒ gof(x)= 3(x⁶ + 512 + 24x⁴ + 192x²) + 1

⇒ gof(x)= 3x⁶ + 72x⁴ + 576x² + 1537

Similarly, fog(x)=f(g(x))

⇒ fog(x)= f(3x³ + 1)

⇒ fog(x)=(3x³ + 1)² + 8

⇒ fog(x)=(9x⁶ + 1 + 6x³) + 8

⇒ fog(x)=9x⁶ + 6x³ + 9

So, gof(x) = 3x⁶ + 72x⁴ + 576x² + 1537 and fog(x) = 9x⁶ + 6x³ + 9