Question

Let f(x) = sin x and g(x) = ln|x|. If the ranges of the composite functions (f_og) and (g_of) are R₁ and R₂ respectively then 

(a) R₁ = {u: –1 < u < 1}, R₂ = {v:– ∞ < v < 0} 

(b) R₁ = {u: –∞ < u ≤ 1}, R₂ = {v: –1 ≤ v ≤ 0}

(c) R₁ = {u: – 1 < u < 1}, R₂ = {v: –∞ < v < 0} 

(d) R₁ = {u: – 1 ≤ u ≤ 1}, R₂ = {v: –∞ < v ≤ 0}

Answer 1

 Correct option  (d) R₁ = {u: – 1 ≤ u ≤ 1}, R₂ = {v: –∞ < v ≤ 0}

Explanation:

We have (f_og)(x) = f(g(x))

= f(ln|x|) = sin(ln |x|)

Thus, R₁ = {u : – 1 ≤  u ≤  1}

Also, (g_of)(x) = g(f(x)) = g(sin x)= ln|sin (x)|

Since 0 ≤  |sin(x)|, ≤  1 we get,

R₂ = {v: –∞ < v ≤  0}