Question

Give the intervals on which the given function is continuous. $$ g(x)=\sqrt{4-x^{2}} $$

Short answer

The function is continuous on the interval \([-2, 2]\).

Step-by-step solution

Identify the Domain of the Function

The function is given as \( g(x) = \sqrt{4 - x^2} \). For the function to be defined, the expression under the square root must be non-negative: \( 4 - x^2 \geq 0 \). This inequality will help us determine the domain where the function is continuous.

Solve the Inequality

To solve \( 4 - x^2 \geq 0 \), rearrange it to \( x^2 \leq 4 \). This implies that \( -2 \leq x \leq 2 \). The endpoints are included because the inequality is non-strict (\( \leq \)).

Establish the Interval of Continuity

A function involving a square root is continuous on its domain where the expression inside the square root is non-negative. Thus, based on the inequality \( -2 \leq x \leq 2 \), the function \( g(x) \) is continuous throughout this interval.

Write the Interval of Continuity

The function \( g(x) = \sqrt{4 - x^2} \) is continuous on the closed interval \([-2, 2]\).

Key concepts

Interval of Continuity

Continuous functions are an essential topic in calculus as they describe functions that have no breaks, jumps or holes in their graphs within certain intervals. When discussing the **interval of continuity**, we refer to the precise range of the input values over which a function consistently behaves as continuous. For the function \( g(x) = \sqrt{4 - x^2} \), it is essential to recognize that **continuity** occurs whenever the function is defined without any interruptions. Since this function includes a square root, it must be continuous wherever \(4 - x^2\) is non-negative.In practical terms, this means identifying the interval where the values of \(x\) allow the function to produce a real number output, as opposed to yielding a complex or undefined result. In this case, we solved the inequality \(4 - x^2 \geq 0\), concluding that the function \(g(x)\) is continuous on the closed interval \([-2, 2]\). This closed interval signifies that the endpoints are included, ensuring continuity at every number between and including \(-2\) and \(2\). Understanding the interval of continuity is critical because it allows you to predict where a function can safely be used without encountering undefined values. When a function behaves continuously over an interval, you can seamlessly transition through that range without interruption, making calculations and predictions about the function's behavior reliable and straightforward.

Domain of a Function

The **domain of a function** refers to all the possible input values (x-values) for which the function is defined. Identifying the domain is crucial because it sets the groundwork for understanding where the function operates without errors or undefined values.For the square root function in our exercise, \( g(x) = \sqrt{4 - x^2} \), identifying the domain requires ensuring that the expression under the square root, \(4 - x^2\), stays non-negative. This means solving the inequality \(4 - x^2 \geq 0\), which implies that the square root is only defined between certain values of \( x \).Upon solving, we find that \( x^2 \leq 4 \), or equivalently, \(-2 \leq x \leq 2 \). This implies the domain of the function \( g(x) = \sqrt{4 - x^2} \) is the interval \([-2, 2]\). This domain covers all x-values where the function produces real output and does not venture into undefined or non-real territories.Thus, the domain helps in identifying where a function is applicable and functional. Knowing the domain is one of the first steps in mathematical analysis, as it informs you about the "space" upon which you can reliably work with the function.

Square Root Function

Square root functions, like the one in the example, \( g(x) = \sqrt{4 - x^2} \), present unique characteristics and requirements for their domain and continuity. They are defined only when the expression inside the square root is non-negative since the square root of a negative number is not defined in the set of real numbers.The square root function \( \sqrt{4 - x^2} \) is a specific type of function known as a radical function, owing to the radical (square root) sign. It inherently restricts its domain to ensure that values remain within the real numbers, avoiding complex numbers unless extended into the complex plane purposely.For such functions:

• The expression under the root determines all behaviors of the function.
• They are continuous in their domain, meaning no breaks or jumps exist within that range.
• The graph of a square root function \( \sqrt{a - x^2} \), like ours, forms a half-circle or an arc on the coordinate plane because it maps values symmetrically around the central axis, given by the square under the root.

Understanding these principles of square root functions allows us to handle them more effectively in calculus and analysis, especially when considering real-world problems or more complex mathematical scenarios.