Problem 24 - Calculus Single Variable

Question

Plot several points, and sketch the graph of the function defined by the given expression. $$ \left\\{\begin{aligned} x^{2}-4 & \text { if } x<2 \\ x+2 & \text { if }-2 \leq x<2 \\ x^{2} & \text { if } 2 \leq x \end{aligned}\right. $$

Step-by-Step Solution

Answer

Graph the function using points from each piece-wise interval, connecting smoothly and addressing discontinuities.

Step 1: Understand Function Definition

The function is piece-wise defined, meaning it has different expressions for different intervals of the variable \(x\). For \(x < 2\), the function is \(x^2 - 4\). For \(-2 \leq x < 2\), the function is \(x + 2\). For \(x \geq 2\), the function is \(x^2\).

Step 2: Evaluate the First Interval \(x < 2\)

Let's evaluate and plot points for the expression \(x^2 - 4\) when \(x < 2\). For instance:- When \(x = -3\), \(f(x) = (-3)^2 - 4 = 9 - 4 = 5\)- When \(x = -1\), \(f(x) = (-1)^2 - 4 = 1 - 4 = -3\)- When \(x = 0\), \(f(x) = 0^2 - 4 = -4\)Plot these points on a graph.

Step 3: Evaluate the Second Interval \(-2 \leq x < 2\)

For \(-2 \leq x < 2\), evaluate the expression \(x + 2\). Consider:- When \(x = -2\), \(f(x) = -2 + 2 = 0\)- When \(x = 0\), \(f(x) = 0 + 2 = 2\)- When \(x = 1\), \(f(x) = 1 + 2 = 3\)Plot these points, ensuring continuity where the intervals meet.

Step 4: Evaluate the Third Interval \(x \geq 2\)

For \(x \geq 2\), use the expression \(x^2\). Evaluate:- When \(x = 2\), \(f(x) = 2^2 = 4\)- When \(x = 3\), \(f(x) = 3^2 = 9\)- When \(x = 4\), \(f(x) = 4^2 = 16\)Plot these points on the graph.

Step 5: Sketch the Graph

Using the points plotted from the three intervals, connect the points smoothly to sketch the graph. Make sure discontinuities at \(x = 2\) transition correctly from one piece to the next, and the open and closed circles are correctly placed at boundaries.

Key Concepts

Graphing Functions

Graphing functions involves visually representing the relationship between input values and their corresponding outputs on a coordinate plane. For piecewise functions like the one given, graphing becomes slightly more complex, as the function is defined by different expressions over different intervals. To graph a piecewise function, you'll need to consider each part separately and plot points accordingly. First, pick a variety of values for the input (x) within each interval defined by the function. Then calculate the outputs (f(x)) using the specified expression for each interval. As you plot these points, be mindful of the conditions that define each piece of the function. This careful plotting will help you successfully connect the points to sketch the entire graph, illustrating how the function behaves across its domain.

Function Continuity

A function is continuous over an interval if there are no breaks, jumps, or holes in the graph within that interval. For the piecewise function we are dealing with, it's important to check the transitions at the boundaries of each interval. These transitions are: from the first to the second interval at the point just below 2, and from the second to the third interval starting at 2.

• For the transition at x = 2, where the second expression ends and the third begins, continuity can be checked by seeing if both the end of one interval and the beginning of the next meet at the same point on the y-axis.
• If they meet at the same y value, the function is continuous at that point; otherwise, it shows a jump or break in continuity.

Looking deeper into our function, you would see how the graph shifts between different segments and whether adjustments are needed to align them neatly at these boundaries.

Evaluating Functions

Evaluating a function means substituting values of x into the function's equation to find the corresponding y values or f(x). This is a crucial step in plotting the graph, as it provides the necessary data points. For the piecewise function, you evaluate each part separately based on the defined intervals.

• In the interval where x < 2, substitute the values of x into the expression x² - 4 and calculate the result.
• In the interval starting from -2 and going up to just below 2, use x + 2 to find f(x).
• For x ≥ 2, the expression x² determines the function's output.

By systematically evaluating these different parts, you gather a collection of points that you will plot when building the graph, showcasing how the function behaves differently across these intervals.

Plotting Points

Plotting points is a straightforward, yet essential part of building graphs for functions. Start by drawing a coordinate plane with the x-axis and y-axis clearly marked. Use the points obtained from evaluating the function to place dots at the corresponding (x, y) coordinates.

• Double-check that you're plotting each x value in its appropriate interval.
• Use an open circle for points where the function is not defined or not equal to the value at the endpoint, and a closed circle where it is defined.
• When all points within each interval are plotted, connect them smoothly. This will often involve drawing straight lines or curves that best represent the equation defined in that section.

Remember, plotting is not just about dots but about understanding how these dots connect to reveal the overall shape of the function's graph.