
Let\[f(x) = \sin x\]and \[g(x) = {\log _e}\left| x \right|\], If the ranges of the composition function \[fog\]and\[gof\]are \[{R_1}\]and\[{R_2}\]respectively, then
A. \[{R_1}\]=\[\left\{ {u: - 1 \le u < 1} \right\}\], \[{R_2}\]=\[\left\{ {v: - \infty < v < 0} \right\}\]
B. \[{R_1}\]=\[\left\{ {u: - \infty < u < 0} \right\}\],\[{R_2} = \left\{ {v: - \infty < v < 0} \right\}\]
C. \[{R_1}\]=\[\left\{ {u: - 1 < u < 1} \right\}\], \[{R_2} = \left\{ {v: - \infty < v < 0} \right\}\]
D. \[{R_1}\]=\[\left\{ {u: - 1 \le u \le 1} \right\}\],\[{R_2} = \left\{ {v: - \infty < v \le 0} \right\}\]
Answer
161.1k+ views
Hint: When we put the value of \[g(x)\]in \[f(x)\]we get \[fog\]and when we put the value of\[f(x)\]in \[g(x)\]we get \[gof\]
Complete step by step solution: Given that \[f(x) = \sin x\]and \[g(x) = {\log _e}\left| x \right|\], Now find \[fog\]and\[gof\]with domain and range
1st \[fog\]=\[f(gx) = \sin (gx) = \sin ({\log _e}\left| x \right|)\]
We know \[\left| x \right| \in R\], So \[{\log _e}\left| x \right| \in R\]
And \[\sin x\]vary from – 1 to 1, So for \[fog\]range \[{R_1}\]=\[\left[ { - 1,1} \right]\] --------(1)
Now, 2nd \[gof\]=\[g(fx) = g(\sin x) = {\log _e}\left| {\sin x} \right|\]
We know the value of \[\left| {\sin x} \right|\]vary from 0 to 1, So the range of \[{\log _e}\left| {\sin x} \right| = - \infty < {\log _e}\left| {\sin x} \right| \le 0\]
Hence range of \[{R_2}\]=\[{\log _e}\left| {\sin x} \right| = \left[ { - \infty ,0} \right]\] ------(2)
Taking both the equation we get \[{R_1}\]=\[\left\{ {u: - 1 \le u \le 1} \right\}\],\[{R_2} = \left\{ {v: - \infty < v \le 0} \right\}\]
Thus, Option (D) is correct.
Note:most of the student Students put wrong values to find composition function like andbecause of misunderstanding of the functions. So student must know the ranges of basic functions like modulus, linear, trigonometric functions, and understanding of composition function.
Complete step by step solution: Given that \[f(x) = \sin x\]and \[g(x) = {\log _e}\left| x \right|\], Now find \[fog\]and\[gof\]with domain and range
1st \[fog\]=\[f(gx) = \sin (gx) = \sin ({\log _e}\left| x \right|)\]
We know \[\left| x \right| \in R\], So \[{\log _e}\left| x \right| \in R\]
And \[\sin x\]vary from – 1 to 1, So for \[fog\]range \[{R_1}\]=\[\left[ { - 1,1} \right]\] --------(1)
Now, 2nd \[gof\]=\[g(fx) = g(\sin x) = {\log _e}\left| {\sin x} \right|\]
We know the value of \[\left| {\sin x} \right|\]vary from 0 to 1, So the range of \[{\log _e}\left| {\sin x} \right| = - \infty < {\log _e}\left| {\sin x} \right| \le 0\]
Hence range of \[{R_2}\]=\[{\log _e}\left| {\sin x} \right| = \left[ { - \infty ,0} \right]\] ------(2)
Taking both the equation we get \[{R_1}\]=\[\left\{ {u: - 1 \le u \le 1} \right\}\],\[{R_2} = \left\{ {v: - \infty < v \le 0} \right\}\]
Thus, Option (D) is correct.
Note:most of the student Students put wrong values to find composition function like andbecause of misunderstanding of the functions. So student must know the ranges of basic functions like modulus, linear, trigonometric functions, and understanding of composition function.
Recently Updated Pages
If tan 1y tan 1x + tan 1left frac2x1 x2 right where x frac1sqrt 3 Then the value of y is

Geometry of Complex Numbers – Topics, Reception, Audience and Related Readings

JEE Main 2021 July 25 Shift 1 Question Paper with Answer Key

JEE Main 2021 July 22 Shift 2 Question Paper with Answer Key

JEE Electricity and Magnetism Important Concepts and Tips for Exam Preparation

JEE Energetics Important Concepts and Tips for Exam Preparation

Trending doubts
JEE Main 2025 Session 2: Application Form (Out), Exam Dates (Released), Eligibility, & More

JEE Main 2025: Derivation of Equation of Trajectory in Physics

Electric Field Due to Uniformly Charged Ring for JEE Main 2025 - Formula and Derivation

Displacement-Time Graph and Velocity-Time Graph for JEE

JEE Main 2026 Syllabus PDF - Download Paper 1 and 2 Syllabus by NTA

JEE Main Eligibility Criteria 2025

Other Pages
JEE Advanced Marks vs Ranks 2025: Understanding Category-wise Qualifying Marks and Previous Year Cut-offs

JEE Advanced 2025: Dates, Registration, Syllabus, Eligibility Criteria and More

JEE Advanced Weightage 2025 Chapter-Wise for Physics, Maths and Chemistry

Degree of Dissociation and Its Formula With Solved Example for JEE

Free Radical Substitution Mechanism of Alkanes for JEE Main 2025

JEE Advanced 2025 Notes
