Problem 6 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

The largest interval lying in \(\left(\frac{-\pi}{2}, \frac{\pi}{2}\right)\) for which the function, \(f(x)=4^{-x^{2}}+\cos ^{-1}\left(\frac{x}{2}-1\right)+\log (\cos x)\), is defined, is (a) \(\left[-\frac{\pi}{4}, \frac{\pi}{2}\right)\) (b) \(\left[0, \frac{\pi}{2}\right)\) (c) \([0, \pi]\) (d) \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\)

Step-by-Step Solution

Answer

Choice (b) \([0, \frac{\pi}{2})\) is the largest interval for which the function is defined.

Step 1: Determine Domain for Exponential Component

For the term \(4^{-x^2}\), there's no restriction since the exponential function \(4^x\) is defined for all real numbers. Thus, \(-\frac{\pi}{2} < x < \frac{\pi}{2}\).

Step 2: Determine Domain for Inverse Cosine Component

For \(\cos^{-1}\left(\frac{x}{2} - 1\right)\) to be defined, the argument \(\frac{x}{2} - 1\) must be within \([-1, 1]\). Solving, \(\frac{x}{2} - 1 = -1\) gives \(x = 0\) and \(\frac{x}{2} - 1 = 1\) gives \(x = 4\). So, \(0 \leq x \leq 4\).

Step 3: Determine Domain for Logarithmic Component

For \(\log(\cos x)\) to be defined, \(\cos x > 0\) must hold. Since \(\cos x > 0\) for \(-\frac{\pi}{2} < x < \frac{\pi}{2}\), it additionally restricts the function depending on where \(\cos x\) is strictly positive.

Step 4: Combine Restrictions

Combine domains from Steps 1, 2, and 3. While Step 1 restricts \(-\frac{\pi}{2} < x < \frac{\pi}{2}\), Step 2 restricts to \(0 \leq x \leq 4\). However, since \(0 \leq x < \frac{\pi}{2}\) from Step 3, the overall restriction confines us to \(0 \leq x < \frac{\pi}{2}\).

Step 5: Determine Largest Interval

Considering the overall restriction as \(0 \leq x < \frac{\pi}{2}\), the largest interval (considering the choices given) matches with choice (b) \(\left[0, \frac{\pi}{2}\right)\).

Key Concepts

Exponential Function

An exponential function is one where the variable is in the exponent. For instance, in our exercise, we have the expression \(4^{-x^2}\). Exponential functions are powerful because they grow very rapidly. However, for this specific term, there's no restriction on the variable \(x\) in terms of domain. This is because any real number input into an exponential function gives back a valid output. Thus, \(-\frac{\pi}{2} < x < \frac{\pi}{2}\) for this part of the function. Exponential components generally do not limit the domain within the realm of real numbers.

Inverse Trigonometric Function

Inverse trigonometric functions, like \(\cos^{-1}\), have specific domain restrictions. Here, we consider \(\cos^{-1}\left(\frac{x}{2} - 1\right)\). The expression inside the inverse cosine should lie between \(-1\) and \(1\). This is because the cosine of an angle can only return values within this interval. When you solve \(\frac{x}{2} - 1\leq 1\) and \(\frac{x}{2} - 1\geq -1\), you get \(0 \leq x \leq 4\). However, because of other restrictions in the function, not every value here may work in the larger domain of the issue.

Logarithmic Function

Logarithmic functions, such as \(\log(\cos x)\), define restrictions based on positivity. Specifically, the log function is only defined for positive inputs. Hence, \(\cos x\) must be greater than zero. This occurs when \(-\frac{\pi}{2} < x < \frac{\pi}{2}\), because the cosine function is positive in this interval. Thus, our real domain for the logarithmic component aligns well with the restrictions around exponential functions, but must still intersect with restrictions from the inverse trigonometric component to find the overall domain.

Interval Restriction

Interval restriction is about finding the overlap between domains of different function components. Here, our function has limitations from the inverse trigonometric and logarithmic parts. By combining these, the domain becomes more refined. From the exponential and logarithmic parts, we know the range should be from \(-\frac{\pi}{2} < x < \frac{\pi}{2}\). However, the inverse cosine limits \(x\) to between \(0\) and \(4\). Thus, after evaluating all pieces, the valid interval is \(0 \leq x < \frac{\pi}{2}\). This is the largest interval satisfying all conditions.

Real Numbers

Real numbers encompass every number that can be thought of between negative and positive infinity, incorporating fractions and irrational numbers as well. When discussing the domains of our functions, we stick to real numbers since they include all possible inputs for the components involved. Exponential, inverse trigonometric, and logarithmic functions we encounter here interact across these real numbers, helping us find where they are all valid. The domain generally needs to be expressed in real number intervals, which are significant when defining comprehensive ranges for any function we tackle.