Problem 45 - Algebra and Trigonometry

Question

Sketch the graph of the piecewise defined function. $$ f(x)=\left\\{\begin{array}{ll}{2} & {\text { if } x \leq-1} \\ {x^{2}} & {\text { if } x>-1}\end{array}\right. $$

Step-by-Step Solution

Answer

The graph has a horizontal line at \( y=2 \) for \( x \leq -1 \) and a parabola \( y=x^2 \) for \( x > -1 \), starting with an open circle at \( (-1, 1) \).

Step 1: Analyze the conditions

The function is defined as a piecewise function with two conditions. The value of the function is \( f(x) = 2 \) when \( x \leq -1 \) and \( f(x) = x^2 \) when \( x > -1 \). We need to sketch the graph considering these two parts.

Step 2: Sketch the part for \( x \leq -1 \)

For \( x \leq -1 \), the function is constant: \( f(x) = 2 \). This means that the graph will be a horizontal line at \( y = 2 \) starting from the line \( x = -1 \) and extending to the left.

Step 3: Sketch the part for \( x > -1 \)

For \( x > -1 \), the function follows \( f(x) = x^2 \), which is the graph of a parabola opening upwards. However, it starts only after \( x = -1 \). The point \( (-1,1) \) is not included, and from there, the parabola begins.

Step 4: Analyze the transition

At \( x = -1 \), the switch happens. The function value is defined by the constant part, \( f(x) = 2 \). However, the quadratic part, \( f(x) = x^2 \), approaches but does not include this value, touching up on the point \( (-1, 1) \). Consider a filled circle at \( (-1, 2) \) and an open circle at \( (-1, 1) \) for the quadratic part.

Step 5: Complete the sketch

Combine both parts on the graph. Draw a horizontal line from \( (-1, 2) \) to the left with a filled circle at \( (-1, 2) \). Then, draw the upward segment of the parabola to the right of \( x = -1 \) with an open circle at \( (-1, 1) \) and the rest of the parabola following the \( y = x^2 \) pattern for \( x > -1 \).

Key Concepts

Piecewise Functions

Piecewise functions are a special type of function that allow us to define a mathematical rule based on different conditions or intervals of the input variable. They are written in a format where each part of the function is given for a specific domain. This enables us to piece together different function rules, essentially "stitching" them into one single function. For example, for the function— \[f(x)=\left\{\begin{array}{ll}{2} & {\text { if } x \leq-1} \ {x^{2}} & {\text { if } x>-1}\end{array}\right. \]- There are two pieces or "rules": - From \(x \leq -1\), the rule is \(f(x) = 2\). - From \(x > -1\), the rule changes to \(f(x) = x^2\).
Piecewise functions are incredibly useful in situations where different rules apply to different scenarios, such as tax brackets, shipping rates, and even physics problems where conditions change. Understanding how to define and interpret these pieces is key to analyzing piecewise functions effectively.

Graph Sketching

Graph sketching is an essential skill that helps visualize functions, including piecewise functions. When sketching a piecewise function graph, you need to consider the behavior of each segment individually and then how they connect together. Here’s a simplified approach to sketching piecewise graphs like the given function:- **Analyze Each Segment**: Start by understanding what each part of the function does. For our example, recognize that for \(x \leq -1\), the graph is a horizontal line at \(y = 2\). For \(x > -1\), it’s a parabola (\(y = x^2\)) starting immediately after \(x = -1\).- **Boundary Points**: These are points where the function switches from one rule to another. Pay attention to these transitions. In graphical terms, you might have an open or closed circle at these points to show whether the point is included in the graph. For \(x = -1\), the graph has a closed circle at \((-1, 2)\) and an open circle at \((-1, 1)\).- **Combine Segments**: Finally, draw each segment on a single coordinate plane, ensuring continuity where appropriate. This is where your open and closed circles will help clarify how to transition between segments.
This structured process not only ensures accuracy in the sketching but also reinforces the understanding of function behavior across its full domain.

Functions and Graphs

Understanding the relationship between functions and their graphs is fundamental for analyzing mathematical ideas. A function can be seen as a rule that assigns every input exactly one output. The graph of a function is a visual representation of all possible inputs (x-values) and their corresponding outputs (y-values). Here’s how to interpret and draw them: - **Function Rules**: The rule of the function dictates the pattern or shape of the graph. In piecewise functions, multiple rules will create combinations of shapes, such as lines and curves. - **Graph Characteristics**: Key features to look for in function graphs include: - Intersections: Points where the graph crosses the x-axis or y-axis. - Vertex or Corner Points: Especially in piecewise functions, where the rule changes. - Continuity: Whether the graph is a smooth continuous line, or if there are jumps or breaks. - **Visual Clarity**: Graphs not only provide a visual understanding but make it easier to identify properties like increasing or decreasing intervals, symmetry, and other behaviors.
By mastering the interplay between functions and their graphs, students can more deeply analyze and predict the behavior of complex mathematical scenarios. They become capable of visualizing abstract concepts, making it a crucial skill in both academics and practical applications.