Problem 13 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

\(f(x)=\sin ^{-1} \log _{2}\left(\frac{1}{2} x^{2}\right)\)

Step-by-Step Solution

Answer

The function \(f(x)=\sin ^{-1}\log _{2}\left(\frac{1}{2} x^{2}\right)\) is evaluated by first squaring the input, then dividing by 2, taking the base-2 logarithm of the result, and finally applying the inverse sine function to the output of the logarithm. This is the result of the successive stages of computation conducted by the function.

Step 1: Identify Function Stages

It is important to recognize the stages in the function. The inside consists of \(x^2\), which is then divided by 2. This output is then used as the input to the \(\log_{2}\) function. The result of this logarithmic function is then input to the \(\sin^{-1}\) or inverse sine function.

Step 2: Input

As the function's input, \(x\) undergoes each stage. Starting with \(x^2\) which will output \(x^2\).

Step 3: Division

Take the output from the previous step (\(x^2\)) and divide it by 2. This results in \(\frac{x^2}{2}\).

Step 4: Logarithm

Apply the logarithmic function to the previous output. Remember, this is a base-2 logarithm. So, we apply the \(\log_{2}\) operator to \(\frac{x^2}{2}\), which gives us \(\log_{2}\left(\frac{x^2}{2}\right)\).

Step 5: Inverse Sine

The output of the previous step, \(\log_{2}\left(\frac{x^2}{2}\right)\), is input into the \(\sin^{-1}\) function. We apply the inverse sine function to get \(\sin^{-1}\left(\log_{2}\left(\frac{x^2}{2}\right)\right)\). This is the final form of our function, \(f(x)\).

Key Concepts

Logarithmic Functions

Logarithmic functions are a powerful tool in mathematics, often used for their unique ability to reverse exponential growth. In our problem, we specifically deal with the logarithm base 2, denoted as \( \log_{2} \). When we take \( \log_{2} \) of a number, we are essentially finding to which power we must raise 2 to get that number. For instance, \( \log_{2}(8) = 3 \) because \( 2^3 = 8 \).

This function is particularly useful for rescaling large ranges of values, as it compresses them into a more manageable scale. In our exercise, we apply the base-2 logarithm to \( \frac{x^2}{2} \). This means that for each value of \( x \), we first square it, divide by 2, and then find the power of 2 that results in this number. This application of logarithms is crucial in function composition as it serves as an intermediate transformation before applying further functions.

Function Composition

Function composition is a method where we apply one function to the results of another. It helps us perform complex operations by breaking them down into simpler, understandable steps. In the problem at hand, we have a composition of three functions: squaring the input \( x \), taking the logarithm, and applying the inverse sine.

• First, start by computing \( x^2 \), which prepares the input for further processing.
• Second, transform \( x^2 \) by dividing by 2. This ensures our input to the logarithm function is correct.
• Third, apply the logarithmic function \( \log_{2} \) to get a reduced, more digestible figure.
• Finally, the transformed input is passed through the inverse sine function \( \sin^{-1} \), which helps us interpret the logarithmic output as an angle.

These steps outline how function composition enables us to extract a meaningful interpretation from a seemingly complex mathematical formula. It emphasizes the importance of each function and their sequence.

Differentiation Techniques

Differentiation is a key concept in calculus that deals with how functions change. When we need to find the derivative of a composition of functions, like in our exercise, we use the chain rule.

The chain rule states that the derivative of a composite function \( f(g(x)) \) is \( f'(g(x)) \cdot g'(x) \). Essentially, we differentiate the outer function first and multiply by the derivative of the inner function.

For the exercise,

• First, differentiate the outermost function, \( \sin^{-1}(u) \), where \( u = \log_{2}(\frac{x^2}{2}) \), which results in \( \frac{1}{\sqrt{1-u^2}} \).
• Then, find the derivative of the logarithmic function applied to \( \frac{x^2}{2} \). Here, the differentiation involves \( \frac{1}{x^2 \ln 2} \times x \) (since the derivative of \( \log_{a}(x) \) is \( \frac{1}{x \ln a} \)), considering the full chain effect.
• Multiply these derivatives as per the chain rule, adjusting each step to account for the next, leading to the complete differentiated form of the original composite function.

These techniques allow for the precise calculation of rates of change, enabling robust mathematical analysis and providing deeper insights into the behavior of complex functions.