Problem 57 - Algebra and Trigonometry Real Mathematics, Real People

Question

Sketch the graph of the piecewise-defined function by hand. $$f(x)=\left\\{\begin{array}{ll} \sqrt{4+x}, & x<0 \\ \sqrt{4-x}, & x \geq 0 \end{array}\right.$$

Step-by-Step Solution

Answer

The graph of the function \(f(x)\) is a piecewise function that is half of an upward-opening parabola from (-4, 0) to (0, 2), and half of a downward-opening parabola from (0, 2) to (4, 0). It is continuously increasing before the point (0, 2) and continuously decreasing after this point.

Step 1: Analyze the function for \(x < 0\)

For \(x < 0\), the function is \(f(x) = \sqrt{4+x}\). Recall that the square root of a number is always non-negative. Therefore, this part of the function is only defined for \(x \geq -4\). For \(x = -4\), we get the point (-4, 0) which will be included in the plot. When \(x\) increases from -4 to 0, \(4 + x\) increases from 0 to 4, so \(\sqrt{4 + x}\) increases from 0 to 2. This means we have a half of an upward-opening parabola shifted horizontally to the left by 4 units when plotting the function for \(x < 0\).

Step 2: Analyze the function for \(x \geq 0\)

For \(x \geq 0\), the function is \(f(x) = \sqrt{4-x}\). This part of the function is only defined for \(x \leq 4\). Relatively to step 1, here the function decreases as \(x\) increases. For \(x = 0\), we get the point (0, 2), and for \(x = 4\), we get the point (4, 0), which will be included in the plot. This means we have the other half of an upward-opening parabola shifted horizontally to the right by 4 units when plotting the function for \(x \geq 0\).

Step 3: Combine the sketches

Now that we have sketched each part of the function on its own, it's time to combine them into one graph. This graph should show a piecewise function that starts at (-4, 0), then follows an upward-opening half-parabola to the point (0, 2), and then follows a downward-opening half-parabola back down to the point (4, 0). Make sure to indicate clearly that the point (0, 2) belongs to the function for \(x \geq 0\).

Key Concepts

Piecewise Functions

Piecewise functions, like the one presented in our exercise, are functions defined by multiple sub-functions, each applying to a certain interval of the input value, or x. They are essential in modeling situations where a rule changes based on the input value. For instance, think of a shipping cost that varies depending on weight or a taxi fare that changes after a certain distance.

To sketch a piecewise function, it is important to consider the domain of each sub-function—where each part starts and ends—alongside the behavior of the function within those intervals. To improve your exercise experience with piecewise functions, visualize each segment separately and remember to include any endpoints or corners where the function rules change, as these are key to understanding the function's overall behavior.

Graphing Functions

Graphing functions is a foundational skill in algebra and mathematics as a whole. It's the process of plotting the set of all points that satisfy the function onto a coordinate plane. A well-drawn graph provides a visual understanding of the function's behavior, such as its increasing and decreasing intervals, intercepts with axes, and any asymptotes.

With piecewise functions, you must graph each segment within its specific domain. This means paying close attention to not just where each segment begins and ends, but also how these segments join together (or don't) at the boundaries. After sketching individual parts, always double-check the points where the rules change to ensure continuity or correctly indicate any jumps or discontinuities in the function.

Square Root Functions

Square root functions are a type of radical function that feature a variable under a square root. These functions generally take the form of f(x) = √x, and their graphs are typically half-parabolas that open upwards—reflecting the non-negative results of a square root operation. However, the domain of these functions is restricted to non-negative numbers, as the square root of a negative number is not real (in the context of real-number calculus).

To graph a square root function, start from the lowest x-value within the domain that makes the number under the square root non-negative and plot the corresponding points as x increases. Adjustments can be made for functions with added constants, such as the example in our exercise, which affects the starting point and orientation of the graph.

Analyzing Functions

Analyzing functions involves looking at their properties to understand their behavior across different intervals and at various points. Key components include the domain and range, intercepts, intervals of increase and decrease, maximums and minimums, continuity, and points of discontinuity.

For the piecewise function in our exercise, analysis involves understanding how the function changes as x crosses the threshold from negative to positive. This involves determining the domain for each piece of the function as well as key points like where each piece starts and ends. By analyzing these aspects, we know how to correctly graph the function and anticipate its behavior throughout its entire domain.