Problem 42 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

Graph a downward parabola for \(x \leq 2\), and a line \(y=x\) for \(x > 2\), with a jump from \(-3\) to \(2\) at \(x=2\).

Step 1: Understand the Function Definition

The function is defined piecewise, meaning it has two distinct expressions based on the value of \(x\). For \(x \leq 2\), the function is defined as \(f(x) = 1 - x^2\). For \(x > 2\), the function is defined as \(f(x) = x\). Let's graph each part separately.

Step 2: Graph the First Piece

The first part of the function is \(f(x) = 1 - x^2\), a downward-opening parabola, defined for \(x \leq 2\). The vertex of the parabola is at \(x = 0\), \(f(0) = 1\). Plotting points: \((0, 1)\), \( (1, 0)\), and \( (2, -3)\). Connect these points to form a curve, stopping at \(x=2\), drawing a filled circle because the value at \(x=2\) is included (since \(x \leq 2\)).

Step 3: Graph the Second Piece

The second part of the function is \(f(x) = x\), a straight line with a slope of 1, starting at \(x = 2\) but not including this point. So \((2, 2)\) is on the line, but we represent it with an open circle to denote it's not included. Continue plotting points, e.g., \((3, 3)\), \((4, 4)\), extending indefinitely upwards and to the right.

Step 4: Combine the Graphs

Combine the plots from Step 2 and Step 3. For \(x \leq 2\), you have the parabola extending to \(x=2\). From \(x = 2\) onwards, the graph continues as a straight line from \((2, 2)\) outwards. Check the continuity at \(x = 2\); here \(f(x)\) transitions from \(-3\) to \(2\), so there is a jump discontinuity.

Key Concepts

Parabolas

Parabolas are ubiquitous in the world of mathematics, characterized as symmetrical, U-shaped curves. A parabola is defined by a quadratic function which typically has the form \( f(x) = ax^2 + bx + c \). In the piecewise function provided, the first part of the function \( f(x) = 1 - x^2 \) is a downward-opening parabola. This specific parabola flips downward because of the negative coefficient in front of \( x^2 \).

The vertex of this parabola, where it reaches its maximum or minimum value, is at the point (0,1). By plotting important points like the vertex and other accessible points \, like \((1, 0)\) and \((2, -3)\), you can draw an accurate representation of the curve. The parabola extends to the left and reaches up to the horizontal line, and due to the restriction \( x \leq 2 \), it stops precisely at this point. Therefore, the parabola part of the graph does not continue to all positive \(x\) values, creating a boundary at \( x = 2 \) with a filled circle, showing inclusion.

Graphing Functions

Graphing functions involves visually representing the relationship between each input and output of a function on a Cartesian coordinate system. For piecewise functions, where different expressions apply in different intervals, graphing can involve a combination of distinct curves or lines.

To graph a piecewise function effectively:

• First, determine the domain for each piece of the function and graph each segment individually.
• For the segment \( f(x) = 1 - x^2 \) when \(x \leq 2\), represent it as a parabola.
• The segment \( f(x) = x \) for \( x > 2 \) is a straightforward linear function, which forms a diagonal line with a slope of 1, where every unit increase in \(x\) results in equal unit increases in \(y\).

Bringing these together in the Cartesian plane, the graph clearly shows how the function behaves differently in each interval. Plotting key points and considering the endpoint inclusions (filled or open circles) help in properly connecting the graph sections.

Discontinuity in Functions

Discontinuities in functions occur where a function abruptly changes its behavior, and they can be seen as breaks or gaps in the graph. In the given piecewise function, there is a jump discontinuity at \( x = 2 \). As the function transitions from the parabola \( f(x) = 1 - x^2 \) to the line \( f(x) = x \), there isn't a smooth passing from one segment to the other.

At \( x = 2 \), the end of the parabola is \( f(2) = -3 \). However, when the line starts from \( x > 2 \), it begins at \( y = 2 \). This difference means the function skips over values between \(-3\) and \(2\), leading to a noticeable jump, marking a classic definition of a jump discontinuity.

To identify discontinuities:

• Observe where the different function pieces connect and see if there's a gap.
• Check if a point fulfills continuity criteria \( f(x) \) when approaching from both directions; if not, expect a discontinuity.
• Jump discontinuities are typically marked by distinct breaks in plotted graphs.

Understanding these concepts solidifies comprehension of more complex behaviors in various functions.