Let f ( x )= sin x and g ( x )= log e | x | . If the ranges of the composition functions fog and gof are R 1 and R 2, respectively, then

Question

Let f ( x )= sin x and g ( x )= log e | x | . If the ranges of the composition functions fog and gof are R 1 and R 2, respectively, then (A) R 1= u :-1 ≤ u <1 R 2= v :-∞ < v < 0  (B) R1= u:-∞ < u < 0 R2= v:-∞ < v < 0  (C) R1= u:-1 < u < 1 R2= v:-∞ < v < 0  (D) R1= u:-1 ≤ u ≤ 1 R2= v:-∞ < v ≤ 0

Tardigrade Admin · Accepted Answer

Q. Let $f (x) = sin x$ and $g (x) = lo g_{e} ∣ x ∣.$ If the ranges of the composition functions fog and gof are $R_{1}$ and $R_{2}$ , respectively, then

$R_{1} = {u : - 1 \leq u < 1}, R_{2} = {v : - \infty < v < 0}$

$R_{1} = {u : - \infty < u < 0}, R_{2} = {v : - \infty < v < 0}$

$R_{1} = {u : - 1 < u < 1}, R_{2} = {v : - \infty < v < 0}$

$R_{1} = {u : - 1 \leq u \leq 1}, R_{2} = {v : - \infty < v \leq 0}$

Q. Let f(x)=sinx and g(x)=loge​∣x∣. If the ranges of the composition functions fog and gof are R1​ and R2​, respectively, then

R1​={u:−1≤u<1},R2​={v:−∞<v<0}

R1​={u:−∞<u<0},R2​={v:−∞<v<0}

R1​={u:−1<u<1},R2​={v:−∞<v<0}

R1​={u:−1≤u≤1},R2​={v:−∞<v≤0}

We have fog (x)=f(g(x))=sin(loge​∣x∣). loge​∣x∣ has range R, for which sin(loge​∣x∣)∈[−1,1] Therefore, R1​={u:−1≤u≤1} Also, gof (x)=g(f(x))=loge​∣sinx∣ ∵0≤∣sinx∣≤1 or −∞<loge​∣sinx∣≤0 or R2​={v:−∞<v≤0}

Q. Let $f (x) = sin x$ and $g (x) = lo g_{e} ∣ x ∣.$ If the ranges of the composition functions fog and gof are $R_{1}$ and $R_{2}$ , respectively, then

$R_{1} = {u : - 1 \leq u < 1}, R_{2} = {v : - \infty < v < 0}$

$R_{1} = {u : - \infty < u < 0}, R_{2} = {v : - \infty < v < 0}$

$R_{1} = {u : - 1 < u < 1}, R_{2} = {v : - \infty < v < 0}$

$R_{1} = {u : - 1 \leq u \leq 1}, R_{2} = {v : - \infty < v \leq 0}$