Question

1) Show that the functions f(x)= x + 1 and g(x)=(x - 1)\are inverse to eachother. ii) Proře by Mathematical induction that 51+2 + 62n+! is divisible by 31 for all natural numbers n. ​

Answer 1

Given, f : R → R and g : R → R

⇒ fog : R → R and gof : R → R (Also, we know that IR : R → R)

So, the domains of all fog, gof and IR are the same.

(fog) (x) = f (g (x)) = f (x−1) = x−1+1= x = IR (x) ... (1)

(gof) (x) = g (f (x)) = g (x+1)= x+1−1 = x=IR (x) ... (2)

From (1) and (2),

(fog) (x) = (gof) (x) = IR (x), ∀x ∈ R

Hence, fog = gof = IR

Answer 2

Answer:

Then P(n) is true for all natural numbers n. 4.2 Solved Examples. Short Answer Type. Prove statements in Examples 1 to 5, ..