Question

Continuity on a closed Interval In Exercises 31-34, discuss the continuity of the function on the closed interval. $$ \begin{array}{ll}{\text { Function }} & {\text { Interval }} \\\ {g(x)=\sqrt{49-x^{2}}} & {[-7,7]}\end{array} $$

Short answer

Because the domain of the function \(g(x) = \sqrt{49-x^2}\) is equal to the given closed interval \([-7,7]\), the function is continuous across this entire interval.

Step-by-step solution

Understanding the Function

The given function is \(g(x) = \sqrt{49-x^{2}}\). This function is a root function and is generally continuous wherever it is defined. Since root functions are undefined for negative radicands, the first step is to figure out when \(49-x^{2}\) will be negative.

Finding the Domain of the Function

To find the domain, you need to solve the inequality \(49 - x^2 \geq 0\). By moving the terms around, you get \(x^{2}\leq 49\). Solving this inequality, you find \(x \in [-7,7]\). This means that the domain of the function is \([-7,7]\).

Comparing the Domain and Given Interval

The domain of \(g(x)\) is equal to the given closed interval of \([-7,7]\). As such, the function is defined everywhere on this interval. Since root functions are always continuous wherever they are defined, \(g(x)\) is continuous on the closed interval \([-7,7]\).

Key concepts

Closed Interval

A closed interval is a set of numbers that includes all the numbers between two endpoints, as well as the endpoints themselves. When we write \([-7, 7]\), this represents a closed interval that includes every number starting from \(-7\) to \(+7\), inclusive of both \(-7\) and \(+7\). This is important in calculus and analysis because it defines the boundaries within which we examine the behavior of a function.

Understanding closed intervals helps in analyzing functions, especially when discussing properties like continuity and differentiability. It ensures that all values at the edges of the interval are considered, thus providing a complete picture of the function's characteristics within that range.

Domain of a Function

The domain of a function is the set of all possible input values (in this case, \(x\)) for which the function is defined. For our function \(g(x) = \sqrt{49 - x^2}\), the domain is determined by considering where the expression under the square root is non-negative.

For the function \(g(x)\), we require \(49 - x^2 \geq 0\). Solving this inequality leads us to \(-7 \leq x \leq 7\), showing that the domain is precisely the closed interval \([-7, 7]\). This means that no value outside this range can be used as an input for the function, ensuring that we avoid undefined or imaginary values from negative radicands.

Inequality Solving

Solving inequalities is a crucial mathematical skill. It involves finding all the values of the variable that make the inequality true. For the given function, we solve \(49 - x^2 \geq 0\) to determine where the function is valid.

This inequality can be rewritten as \(x^2 \leq 49\). Taking square roots on both sides, we reevaluate it as \(|x| \leq 7\). This results in the equivalent expression \(-7 \leq x \leq 7\). Solving inequalities like these helps determine the valid domain of a function, where all mathematical operations within the function remain real and defined.

Continuous Functions

A continuous function is one in which small changes in the input lead to small changes in the output, without any abrupt jumps or breaks. In mathematical terms, a function \(f(x)\) is continuous over an interval if it is defined for every point in that interval, and the limit of \(f(x)\) as \(x\) approaches any point within the interval equals the function’s value at that point.

For \(g(x) = \sqrt{49 - x^2}\), it's stated that it is continuous on \([-7, 7]\), the same interval as its domain. This is because the function is defined and meets the continuous criteria across \([-7, 7]\) without any breaks or interruptions. Continuous functions like these are significant because they ensure predictable and smooth behavior across their defined intervals.