Question

Sketch a graph of the function and find its domain and range. Use a graphing utility to verify your graph. $$ f(x)=\sqrt{9-x^{2}} $$

Short answer

The domain of the function \(f(x) = \sqrt{9-x^{2}}\) is \([-3, 3]\), and the range is \([0, 3]\). This function's graph forms a semicircle above the x-axis, from \(-3\) to \(3\) with a maximum point at \((0, 3)\).

Step-by-step solution

Find the Domain

The domain of \(f(x) = \sqrt{9-x^{2}}\) will be found by setting the expression inside the square root \(9 - x^{2}\) greater than or equal to zero: \[9 - x^{2} \geq 0\]\[-x^{2} \geq -9\]\[x^{2} \leq 9\] Therefore, the domain will be all \(x\) between -3 and 3, including -3 and 3. In interval notation, the domain is \([-3, 3]\).

Find the Range

For given \(x\) values in the domain, the smallest value that \(\sqrt{9 - x^{2}}\) will take on is 0 (which occurs when \(x = -3\) or \(x = 3\)), and the largest value is \(\sqrt{9 - (-3)^{2}} = \sqrt{9} = 3\) (which occurs when \(x = 0\)). So the range of the function will be \(y\) between 0 and 3 (inclusive). In interval notation, the range is \([0, 3]\).

Graphing the Function

Plot the function \(f(x) = \sqrt{9 - x^{2}}\) using a graphing utility. This function will produce a semicircular shape above the x-axis, within the domain \([-3, 3]\). The graph starts at \((-3, 0)\), increases to a maximum at \((0, 3)\), and then decreases back to \(3, 0\). Also, check the graph to verify that the range is indeed \([0, 3]\).

Key concepts

Finding Domain and Range

When dealing with functions in calculus, one of the fundamental skills is to determine their domain and range. The domain of a function is the set of all possible input values (typically representing the x-values) for which the function is defined, and the range is the set of all possible output values (y-values) that the function can produce.

In the case of the function f(x) = \( \sqrt{9-x^{2}} \), the domain is determined by the requirement that the expression under the square root must be non-negative, since we cannot take the square root of a negative number in the set of real numbers. By solving the inequality \(9 - x^{2} \geq 0\), we find that the domain is all x-values between -3 and 3. This includes -3 and 3, as they make the expression inside the square root equal to zero.

Similarly, the range is found by considering the smallest and largest values of the function. Since the smallest value of the square root expression is 0, and it occurs when x equals -3 or 3, and the largest value is 3, which occurs when x is 0, the range of the function is from 0 to 3, inclusive. It's critical for students to understand and practice these steps to find the domain and range of various functions.

Interval Notation

Interval notation is a method of writing sets of numbers that describe the domain or range of a function. This notation is very concise and is favored for its clarity.

For example, let's consider how we express the domain and range from our function f(x) = \( \sqrt{9-x^{2}} \). The domain, which includes all x-values from -3 to 3, is written in interval notation as \( [-3, 3]\). The square brackets indicate that the endpoints are included in the set, known as 'closed' intervals.

On the other hand, if the endpoints were not included, like in a function that cannot equal a certain value, we would use parentheses (or round brackets) to denote that the endpoints are excluded, which are 'open' intervals. Understanding and using the correct symbols is crucial for accurately conveying the sets of values in the domain and range.

Graphing Utilities

Graphing utilities are invaluable tools in understanding the behavior of functions, particularly in calculus. A graphing utility can range from a simple online graphing calculator to more sophisticated software like Desmos or even a hand-held graphing calculator like the TI-84.

These utilities allow students to visually interpret the function, offering a way to confirm the findings for domain and range. After plotting the function \( f(x) = \sqrt{9 - x^{2}} \) on a graphing utility, one can observe the semicircular shape that validates our calculations: it indeed lies within the domain \( [-3, 3] \) and does not exceed the range \( [0, 3] \). This visual confirmation is a crucial step in the learning process, as it cements the understanding of theoretical work with a tangible graph.

Functions in Calculus

In the context of calculus, functions are mathematical entities that describe a relationship between two sets of numbers. The function f(x) = \( \sqrt{9-x^{2}} \) is a classic example used in many calculus textbooks because it involves concepts like roots and even implicit differentiation.

As students progress in calculus, they encounter functions that are more complex. They must not only be adept at finding the domain and range but also at understanding the nuances of how these functions behave. This behavior is studied through limits, derivatives, and integrals, which are essential in analyzing the slopes of tangents (derivatives) or the area under curves (integrals). This particular function, with its semicircular graph, could also segue into discussions on solid revolutions and volumes generated by rotating a function around an axis. A strong foundation in these basic concepts is critical for tackling more advanced calculus problems.