Question

In Problems \(49-54\), determine the largest interval over which the given function is continuous. $$ f(x)=\sqrt{25-x^{2}} $$

Short answer

The largest interval where the function is continuous is [-5, 5].

Step-by-step solution

Identify Function Type

The function given is \( f(x) = \sqrt{25 - x^2} \). This is a square root function.

Determine Domain of Inner Function

To ensure the term \( 25 - x^2 \) is non-negative (because you cannot take the square root of a negative number in the real number system), solve the inequality \( 25 - x^2 \geq 0 \).

Solve the Inequality

Re-arrange and solve \( 25 \geq x^2 \), which implies \( -5 \leq x \leq 5 \) because squaring \( x \) makes sense within these bounds for non-negative values.

Analyze Continuity Within the Interval

The function \( f(x) = \sqrt{25 - x^2} \) is continuous wherever its argument \( 25 - x^2 \) is non-negative. Since \( -5 \leq x \leq 5 \) satisfies this condition, \( f(x) \) is continuous on this interval.

State the Largest Interval

Combine all information to conclude that the largest interval over which \( f(x) \) is continuous is \([-5, 5]\).

Key concepts

Square Root Function

A square root function, such as the one given by \( f(x) = \sqrt{25 - x^2} \), involves finding the square root of a particular expression. These functions are characterized by the presence of a radical symbol, typically extracting the non-negative square root of a number or expression.

Square root functions have a key attribute: they are only defined for non-negative inputs in the set of real numbers. This means the expression inside the square root, called the 'radicand', must be zero or positive. Calculating the square root of a negative number within the real number system is not possible without involving imaginary numbers, which are beyond the scope of basic elementary functions.

In our specific function, \( 25 - x^2 \), this expression must stay non-negative for the function to be defined. Thus, understanding the nature of the square root function is essential for determining its domain.

Domain Analysis

Domain analysis involves finding the set of all possible input values ('x-values') for which a function is defined. For the square root function \( f(x) = \sqrt{25 - x^2} \), the primary focus is ensuring that the inner expression \( 25 - x^2 \) is non-negative.

To find the domain, you need to solve the inequality \( 25 - x^2 \geq 0 \). This analysis determines where \( f(x) \) is not only defined but also behaves as expected based on the rules of real-number arithmetic.

• Identify what makes the radicand non-negative: \( 25 - x^2 \geq 0 \)
• Re-organize to determine the limits for \(x\): \( x^2 \leq 25 \)
• Solve for \(x\) to establish the interval: \( -5 \leq x \leq 5 \)

The domain is thus \(-5\) to \(5\), inclusively, representing all values for which the function's result will be a real number.

Inequality Solving

Solving inequalities like \( 25 - x^2 \geq 0 \) is at the heart of determining the domain of square root functions. It's about understanding both the mathematical manipulation and logical reasoning behind these expressions.

Start by re-arranging the inequality to \( x^2 \leq 25 \). This step is essential, as it uncovers the range of \(x\) values where the radicand is non-negative:

• Consider the critical points where \( x^2 = 25 \), which are \( x = 5 \) and \( x = -5 \).
• Conduct a test on the intervals \((-\infty, -5)\), \([-5, 5]\), and \((5, \infty)\).
• Determine that only values within \([-5, 5]\) satisfy \( x^2 \leq 25 \).

Using this logical process ensures that you correctly determine where the function is defined and continuous.

Continuous Functions

A function is continuous over an interval if there are no breaks, jumps, or holes in its graph within that range. For \( f(x) = \sqrt{25 - x^2} \), the argument \( 25 - x^2 \) being non-negative ensures continuity.

The continuous nature of a function is important for both theoretical understanding and practical applications. Continuity means that small changes in \(x\) lead to small changes in \(f(x)\), providing a reliable and smooth function behavior.

For \(f(x)\), let's analyze where the function stays continuous:

• The function is defined and continuous over \([-5, 5]\) because the radicand \(25 - x^2\) remains non-negative throughout this interval.
• There are no gaps or jumps, making \(f(x)\) smooth and unbroken on this interval.

Recognizing continuity helps ensure accurate predictions and analyses when working with functions.