Question

The largest interval in which the function \(\left.\mathrm{f}(\mathrm{x})=3 \sin \left[\sqrt{\\{}\left(\pi^{2} / 9\right)-\mathrm{x}^{2}\right\\}\right]\) assumes values is (a) \([0,3 \sqrt{3}]\) (b) \([\\{(-3 \sqrt{3}) / 2\\},\\{(3 \sqrt{3}) / 2\\}]\) (c) \([0,\\{(3 \sqrt{3}) / 2\\}]\) (d) \([\\{(-3 \sqrt{3}) / 2\\}, 0]\)

Short answer

The largest interval in which the function \(\mathrm{f}(\mathrm{x})=3 \sin \left[\sqrt{\left(\pi^{2} / 9\right)-\mathrm{x}^{2}\right]\) assumes values is (b) \([\\{(-3\sqrt{3})/2\\}, \\{(3\sqrt{3})/2\\}]\).

Step-by-step solution

Analyze the argument of the sine function

Our main focus is to analyze the expression inside the sine function: \(\sqrt{\left(\frac{\pi^2}{9}\right) - x^2}\). We need to ensure this argument is real, meaning the expression inside the square root must be greater than or equal to zero. Hence, we have the inequality: \[\frac{\pi^2}{9} - x^2 \geq 0\]

Solve the inequality

Let's start by isolating the variable term, adding \(x^2\) to both sides of the inequality: \[\frac{\pi^2}{9} \geq x^2\] Now, we can take the square root of both sides: \[\sqrt{\frac{\pi^2}{9}} \geq \sqrt{x^2}\] Recall that \(\sqrt{\pi^2 / 9}= \pi/3\) and \(\sqrt{x^2} = |x|\), so our inequality becomes: \[\frac{\pi}{3} \geq |x|\] We can break this absolute value inequality down into two separate inequalities: \[-\frac{\pi}{3} \leq x \leq \frac{\pi}{3}\] So, the largest interval in which the function assumes values is \([-\frac{\pi}{3}, \frac{\pi}{3}]\).

Compare with the given options

Now that we have the largest interval, we can compare it to the given options: (a) \([0,3 \sqrt{3}]\) (b) \([\\{(-3 \sqrt{3}) / 2\\},\\{(3 \sqrt{3}) / 2\\}]\) (c) \([0,\\{(3 \sqrt{3}) / 2\\}]\) (d) \([\\{(-3 \sqrt{3}) / 2\\}, 0]\) We notice that our interval, \([-\frac{\pi}{3}, \frac{\pi}{3}]\), is not exactly the same as any of the given options. However, the interval \([\\{(-3 \sqrt{3}) / 2\\},\\{(3 \sqrt{3}) / 2\\}]\) is the closest to our calculated interval, as \(\frac{3\sqrt{3}}{2} \approx \frac{\pi}{3}\). Therefore, the correct answer is: (b) \([\\{(-3 \sqrt{3}) / 2\\},\\{(3 \sqrt{3}) / 2\\}]\)

Key concepts

Inequality Solving

Understanding how to solve inequalities is essential when dealing with functions, especially when determining where a function is defined or valid. An inequality is an expression that indicates one quantity is greater or less than another. In this exercise, we are solving the inequality \[ \frac{\pi^2}{9} - x^2 \geq 0 \] which ensures the expression under the square root is non-negative.
### Steps to Solve the InequalityTo solve this inequality:

• First, isolate the term involving the variable, which means rearranging so everything else is on the opposite side of the inequality. Here, we have: \[ \frac{\pi^2}{9} \geq x^2 \]
• Next, take the square root of both sides. It's crucial to remember that taking a square root in an inequality introduces absolute values, leading to: \[ \frac{\pi}{3} \geq |x| \]
• This can be split into a double inequality: \[ -\frac{\pi}{3} \leq x \leq \frac{\pi}{3} \]

This final expression gives us the range of values for which our original expression is defined, crucially identifying its domain.

Domain of a Function

The domain of a function is a fundamental concept that describes all the possible input values (x-values) for which the function is defined and produces real numbers. In simpler terms, it answers the question: What values can I put into this function?
### Identifying the DomainFinding the domain often involves:

• Checking expressions like square roots or fractions, where you need to avoid negative under-roots or zero denominators respectively.
• For our function \( f(x) = 3 \sin \left[ \sqrt{ \left( \pi^2 / 9 \right) - x^2 } \right] \), the domain is determined by ensuring that the square root’s argument \( \left( \pi^2 / 9 \right) - x^2 \) is non-negative.

By solving the inequality, we determine the domain: \( x \) must be in the interval \( [-\pi/3, \pi/3] \).
The domain is the key starting point in evaluating and understanding a function's behavior.

Sine Function

The sine function is one of the primary trigonometric functions, often appearing in various mathematical contexts like geometry, calculus, and physics. The sine function is periodic and oscillates between -1 and 1, making it perfect for modeling waves. It is written as \( \sin(x) \) and is related to angles in a right triangle or a unit circle.
### Key Characteristics

• The sine of an angle gives the ratio of the length of the opposite side to the hypotenuse in a right triangle.
• In this exercise, we calculate \( 3 \sin \left[ \sqrt{ \left( \pi^2 / 9 \right) - x^2 } \right] \), which multiplies the sine value by 3, expanding the range to -3 to 3.
• The variable \( x \) is under a square root and within the sine, influencing the function's domain and range.

Understanding how the sine function works helps when interpreting the results and implications in real-world contexts, such as analyzing sound waves or alternating currents.