Animated Solution for Mathematics - Functions: Let $f(x)=\sin x$ and $g(x)=\ln |x|$. If the ranges of the composition functions $f\circ g$ and $g\circ f$ are $R_1$ and $R_2$ respectively, then

We are given two functions: $f(x)=\sin x$ and $g(x)=\ln |x|$. Our objective is to determine the ranges $R_1$ and $R_2$ for the composite functions $f\circ g(x)$ and $g\circ f(x)$, respectively.

Understanding these compositions requires careful tracking of how the output of the inner function serves as the input for the outer function.

The composition $f\circ g(x)$ is defined as $f(g(x))$. Here, the inner function $g(x)=\ln |x|$ acts first.

For any $x\in \mathbb{R}\setminus \{0\}$, the absolute value $|x|$ takes values in the interval $(0,\infty )$. Consequently, the range of $g(x)=\ln |x|$ is the entire set of real numbers, $(-\infty ,\infty )$.

Now, we pass this output into the outer function $f(y)=\sin y$. Since the input $y$ covers all real numbers, the sine function oscillates through its full range.

Next, we evaluate $g\circ f(x)=g(f(x))=\ln |\sin x|$. The inner function $f(x)=\sin x$ has a range of $[-1,1]$.

Applying the absolute value, the expression $|\sin x|$ results in a range of $[0,1]$. We must now evaluate the outer function $g(z)=\ln z$ using this interval as the domain.

Crucially, the natural logarithm is only defined for strictly positive values. Since $\ln (0)$ is undefined, we must exclude $0$ from our input domain.

The effective input range for the logarithm becomes $(0,1]$. We analyze the behavior of $\ln z$ over this interval:

Thus, the range of the second composition is:

The Beauty of Constraints

We have successfully determined the ranges:

In JEE Advanced mathematics, the key to success lies in strictly respecting the domain and range constraints of every function. Always verify the validity of the input before passing it to the next stage of a composition.