Problem 26 - Calculus for JEE ADVANCED

Question

\(\cos ^{-1}\left(\frac{2-|x|}{4}\right)\) exists if $$ -1 \leq \frac{2-|x|}{4} \leq 1 $$ or \(\quad-6 \leq-|x| \leq 2\) or \(\quad-2 \leq|x| \leq 6\) or \(\quad|x| \leq 6\) or \(\quad-6 \leq x \leq 6\) The function \([\log (3-x)]^{-1}=\frac{1}{\log (3-x)}\) is defined if \(3-x>0\) and \(x \neq 2\), i.e., if \(x \neq 2\) and \(x<3\). Thus, the domain of the given function is $$ \\{x \mid-6 \leq x \leq 6\\} \cap\\{x \mid x \neq 2, x<3\\}=[-6,2) \cup(2,3) $$

Step-by-Step Solution

Answer

The domain of the function is \([-6, 2) \cup (2, 3)\).

Step 1: Determine the Range of the Expression in Arccosine

The expression \( \cos^{-1}(\frac{2-|x|}{4}) \) exists if the input to the arccosine function is within the range \([-1, 1]\). Start by solving the inequality \( -1 \leq \frac{2-|x|}{4} \leq 1 \).

Step 2: Simplify the Inequality

Multiply the entire inequality \( -1 \leq \frac{2-|x|}{4} \leq 1 \) by 4 to remove the fraction: \(-4 \leq 2 - |x| \leq 4\).

Step 3: Solve for Absolute Value

Subtract 2 from each part of the inequality \(-4 \leq 2 - |x| \leq 4\) to center around \(-|x|\): \(-6 \leq -|x| \leq 2\). Divide by -1, flipping the inequality signs: \(2 \geq |x| \geq -6\), simplifying to \(|x| \leq 6\).

Step 4: Determine the Range of x

The inequality \(|x| \leq 6\) means \(-6 \leq x \leq 6\).

Step 5: Determine Conditions for Logarithm

For \( [\log(3-x)]^{-1} \) to be defined, \( \log(3-x) > 0 \) must hold. This requires \(3-x > 0\), giving \(x < 3\). Also, \(x\) should not equal 2 because division by zero is undefined, hence \(x eq 2\).

Step 6: Find Intersection of Conditions

Combine the conditions: The domain is \(-6 \leq x \leq 6\) (from arccosine) intersected with \(x < 3\) and \(x eq 2\) (from the logarithm part).

Step 7: Express the Domain

The domain is the set of all \(x\) values that satisfy both conditions: \([-6, 2) \cup (2, 3)\).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions, such as the inverse cosine function (\(\cos^{-1}\)), find the angle whose trigonometric function is a given numeric value. These functions are only defined within specific ranges. For the inverse cosine function, the input must lie within \([-1, 1]\). Think of it as the angle within the range of \([0, \pi]\) that produces that cosine value.

• To ensure the expression \(\cos^{-1}(\frac{2-|x|}{4})\) is valid, the inner expression \(\frac{2-|x|}{4}\) must remain within \([-1, 1]\).
• By solving the inequality, we found that \(|x|\) can be up to 6, indicating te range for \(x\) is \[-6, 6\].

Understanding the domain of trigonometric functions helps avoid errors in calculations, like those relating to undefined angles.

Absolute Value Inequalities

Absolute value inequalities involve expressions where the absolute value of a term must be less than or greater than another value. Absolute value \(|x|\) represents the distance of \(x\) from 0 on the number line.

• Given \(-2 \leq |x| \leq 6\), the absolute value is expressed as two linear inequalities: \(-6 \leq x \leq 6\).
• Solving these inequalities provides a range of potential values \(x\) can take.

When tackling problems involving absolute values, always remember that the absolute value can split into two separate conditions to thoroughly consider all potential solutions.

Logarithmic Functions

Logarithmic functions like \(\log(3-x)\) are the inverses of exponential functions. They require the argument to be positive. Thus, a logarithm exists only for values producing positive results; in this instance, \(3-x\) must be greater than zero.

• This converts to the condition \(x < 3\), ensuring valid input.
• Furthermore, \(x eq 2\) prevents division by zero in the expression \([\log(3-x)]^{-1}\).

Logarithms help express exponential relationships in a simple form, and understanding their domains is pivotal in ensuring valid solutions across a variety of problems.